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Metamath Proof Explorer

Theorem usgrexmpl2trifr

Description: G is triangle-free. (Contributed by AV, 10-Aug-2025)

Ref Expression
Hypotheses usgrexmpl2.v ⊢ 𝑉 = ( 0 ... 5 )
usgrexmpl2.e ⊢ 𝐸 = ⟨“ { 0 , 1 } { 1 , 2 } { 2 , 3 } { 3 , 4 } { 4 , 5 } { 0 , 3 } { 0 , 5 } ”⟩
usgrexmpl2.g ⊢ 𝐺 = ⟨ 𝑉 , 𝐸 ⟩
Assertion usgrexmpl2trifr ¬ ∃ 𝑡 𝑡 ∈ ( GrTriangles ‘ 𝐺 )

Proof

Step Hyp Ref Expression
1 usgrexmpl2.v ⊢ 𝑉 = ( 0 ... 5 )
2 usgrexmpl2.e ⊢ 𝐸 = ⟨“ { 0 , 1 } { 1 , 2 } { 2 , 3 } { 3 , 4 } { 4 , 5 } { 0 , 3 } { 0 , 5 } ”⟩
3 usgrexmpl2.g ⊢ 𝐺 = ⟨ 𝑉 , 𝐸 ⟩
4 1 2 3 usgrexmpl2nb0 ⊢ ( 𝐺 NeighbVtx 0 ) = { 1 , 3 , 5 }
5 4 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ↔ 𝑏 ∈ { 1 , 3 , 5 } )
6 vex ⊢ 𝑏 ∈ V
7 6 eltp ⊢ ( 𝑏 ∈ { 1 , 3 , 5 } ↔ ( 𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5 ) )
8 5 7 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ↔ ( 𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5 ) )
9 4 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ↔ 𝑐 ∈ { 1 , 3 , 5 } )
10 vex ⊢ 𝑐 ∈ V
11 10 eltp ⊢ ( 𝑐 ∈ { 1 , 3 , 5 } ↔ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) )
12 9 11 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ↔ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) )
13 eqtr3 ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 1 ) → 𝑏 = 𝑐 )
14 13 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 1 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
15 ax-1ne0 ⊢ 1 ≠ 0
16 neeq1 ⊢ ( 𝑏 = 1 → ( 𝑏 ≠ 0 ↔ 1 ≠ 0 ) )
17 15 16 mpbiri ⊢ ( 𝑏 = 1 → 𝑏 ≠ 0 )
18 17 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑏 ≠ 0 )
19 18 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 0 )
20 19 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
21 3ne0 ⊢ 3 ≠ 0
22 neeq1 ⊢ ( 𝑐 = 3 → ( 𝑐 ≠ 0 ↔ 3 ≠ 0 ) )
23 21 22 mpbiri ⊢ ( 𝑐 = 3 → 𝑐 ≠ 0 )
24 23 adantl ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑐 ≠ 0 )
25 24 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 0 )
26 25 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
27 19 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
28 25 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
29 27 28 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
30 2re ⊢ 2 ∈ ℝ
31 2lt3 ⊢ 2 < 3
32 30 31 gtneii ⊢ 3 ≠ 2
33 neeq1 ⊢ ( 𝑐 = 3 → ( 𝑐 ≠ 2 ↔ 3 ≠ 2 ) )
34 32 33 mpbiri ⊢ ( 𝑐 = 3 → 𝑐 ≠ 2 )
35 34 adantl ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑐 ≠ 2 )
36 35 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 2 )
37 36 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
38 1re ⊢ 1 ∈ ℝ
39 1lt3 ⊢ 1 < 3
40 38 39 gtneii ⊢ 3 ≠ 1
41 neeq1 ⊢ ( 𝑐 = 3 → ( 𝑐 ≠ 1 ↔ 3 ≠ 1 ) )
42 40 41 mpbiri ⊢ ( 𝑐 = 3 → 𝑐 ≠ 1 )
43 42 adantl ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑐 ≠ 1 )
44 43 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 1 )
45 44 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
46 37 45 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
47 1ne2 ⊢ 1 ≠ 2
48 neeq1 ⊢ ( 𝑏 = 1 → ( 𝑏 ≠ 2 ↔ 1 ≠ 2 ) )
49 47 48 mpbiri ⊢ ( 𝑏 = 1 → 𝑏 ≠ 2 )
50 49 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑏 ≠ 2 )
51 50 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 2 )
52 51 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
53 36 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
54 52 53 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
55 29 46 54 3jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
56 38 39 ltneii ⊢ 1 ≠ 3
57 neeq1 ⊢ ( 𝑏 = 1 → ( 𝑏 ≠ 3 ↔ 1 ≠ 3 ) )
58 56 57 mpbiri ⊢ ( 𝑏 = 1 → 𝑏 ≠ 3 )
59 58 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑏 ≠ 3 )
60 59 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 3 )
61 60 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
62 1lt4 ⊢ 1 < 4
63 38 62 ltneii ⊢ 1 ≠ 4
64 neeq1 ⊢ ( 𝑏 = 1 → ( 𝑏 ≠ 4 ↔ 1 ≠ 4 ) )
65 63 64 mpbiri ⊢ ( 𝑏 = 1 → 𝑏 ≠ 4 )
66 65 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑏 ≠ 4 )
67 66 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 4 )
68 67 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
69 61 68 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
70 67 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
71 1lt5 ⊢ 1 < 5
72 38 71 ltneii ⊢ 1 ≠ 5
73 neeq1 ⊢ ( 𝑏 = 1 → ( 𝑏 ≠ 5 ↔ 1 ≠ 5 ) )
74 72 73 mpbiri ⊢ ( 𝑏 = 1 → 𝑏 ≠ 5 )
75 74 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → 𝑏 ≠ 5 )
76 75 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 5 )
77 76 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
78 70 77 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
79 19 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
80 25 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
81 79 80 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
82 69 78 81 3jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
83 55 82 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
84 20 26 83 jca31 ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
85 84 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 3 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
86 17 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑏 ≠ 0 )
87 86 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 0 )
88 87 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
89 58 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑏 ≠ 3 )
90 89 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 3 )
91 90 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
92 87 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
93 0re ⊢ 0 ∈ ℝ
94 5pos ⊢ 0 < 5
95 93 94 gtneii ⊢ 5 ≠ 0
96 neeq1 ⊢ ( 𝑐 = 5 → ( 𝑐 ≠ 0 ↔ 5 ≠ 0 ) )
97 95 96 mpbiri ⊢ ( 𝑐 = 5 → 𝑐 ≠ 0 )
98 97 adantl ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑐 ≠ 0 )
99 98 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 0 )
100 99 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
101 92 100 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
102 2lt5 ⊢ 2 < 5
103 30 102 gtneii ⊢ 5 ≠ 2
104 neeq1 ⊢ ( 𝑐 = 5 → ( 𝑐 ≠ 2 ↔ 5 ≠ 2 ) )
105 103 104 mpbiri ⊢ ( 𝑐 = 5 → 𝑐 ≠ 2 )
106 105 adantl ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑐 ≠ 2 )
107 106 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 2 )
108 107 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
109 49 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑏 ≠ 2 )
110 109 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 2 )
111 110 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
112 108 111 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
113 110 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
114 90 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
115 113 114 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
116 101 112 115 3jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
117 90 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
118 65 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑏 ≠ 4 )
119 118 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 4 )
120 119 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
121 117 120 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
122 119 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
123 74 adantr ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → 𝑏 ≠ 5 )
124 123 neneqd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 5 )
125 124 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
126 122 125 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
127 87 orcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
128 99 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
129 127 128 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
130 121 126 129 3jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
131 116 130 jca ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
132 88 91 131 jca31 ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
133 132 olcd ⊢ ( ( 𝑏 = 1 ∧ 𝑐 = 5 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
134 14 85 133 3jaodan ⊢ ( ( 𝑏 = 1 ∧ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
135 neeq1 ⊢ ( 𝑏 = 3 → ( 𝑏 ≠ 0 ↔ 3 ≠ 0 ) )
136 21 135 mpbiri ⊢ ( 𝑏 = 3 → 𝑏 ≠ 0 )
137 136 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑏 ≠ 0 )
138 137 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 0 )
139 138 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
140 neeq1 ⊢ ( 𝑐 = 1 → ( 𝑐 ≠ 0 ↔ 1 ≠ 0 ) )
141 15 140 mpbiri ⊢ ( 𝑐 = 1 → 𝑐 ≠ 0 )
142 141 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑐 ≠ 0 )
143 142 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 0 )
144 143 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
145 138 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
146 143 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
147 145 146 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
148 58 necon2i ⊢ ( 𝑏 = 3 → 𝑏 ≠ 1 )
149 148 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑏 ≠ 1 )
150 149 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 1 )
151 150 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
152 neeq1 ⊢ ( 𝑏 = 3 → ( 𝑏 ≠ 2 ↔ 3 ≠ 2 ) )
153 32 152 mpbiri ⊢ ( 𝑏 = 3 → 𝑏 ≠ 2 )
154 153 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑏 ≠ 2 )
155 154 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 2 )
156 155 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
157 151 156 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
158 155 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
159 neeq1 ⊢ ( 𝑐 = 1 → ( 𝑐 ≠ 2 ↔ 1 ≠ 2 ) )
160 47 159 mpbiri ⊢ ( 𝑐 = 1 → 𝑐 ≠ 2 )
161 160 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑐 ≠ 2 )
162 161 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 2 )
163 162 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
164 158 163 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
165 147 157 164 3jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
166 neeq1 ⊢ ( 𝑐 = 1 → ( 𝑐 ≠ 4 ↔ 1 ≠ 4 ) )
167 63 166 mpbiri ⊢ ( 𝑐 = 1 → 𝑐 ≠ 4 )
168 167 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑐 ≠ 4 )
169 168 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 4 )
170 169 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
171 42 necon2i ⊢ ( 𝑐 = 1 → 𝑐 ≠ 3 )
172 171 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑐 ≠ 3 )
173 172 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 3 )
174 173 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
175 170 174 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
176 neeq1 ⊢ ( 𝑐 = 1 → ( 𝑐 ≠ 5 ↔ 1 ≠ 5 ) )
177 72 176 mpbiri ⊢ ( 𝑐 = 1 → 𝑐 ≠ 5 )
178 177 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → 𝑐 ≠ 5 )
179 178 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 5 )
180 179 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
181 169 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
182 180 181 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
183 138 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
184 143 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
185 183 184 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
186 175 182 185 3jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
187 165 186 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
188 139 144 187 jca31 ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
189 188 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 1 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
190 eqtr3 ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 3 ) → 𝑏 = 𝑐 )
191 190 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 3 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
192 136 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑏 ≠ 0 )
193 192 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 0 )
194 193 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
195 97 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑐 ≠ 0 )
196 195 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 0 )
197 196 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
198 193 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
199 196 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
200 198 199 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
201 148 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑏 ≠ 1 )
202 201 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 1 )
203 202 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
204 177 necon2i ⊢ ( 𝑐 = 5 → 𝑐 ≠ 1 )
205 204 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑐 ≠ 1 )
206 205 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 1 )
207 206 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
208 203 207 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
209 153 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑏 ≠ 2 )
210 209 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 2 )
211 210 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
212 105 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑐 ≠ 2 )
213 212 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 2 )
214 213 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
215 211 214 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
216 200 208 215 3jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
217 4re ⊢ 4 ∈ ℝ
218 4lt5 ⊢ 4 < 5
219 217 218 gtneii ⊢ 5 ≠ 4
220 neeq1 ⊢ ( 𝑐 = 5 → ( 𝑐 ≠ 4 ↔ 5 ≠ 4 ) )
221 219 220 mpbiri ⊢ ( 𝑐 = 5 → 𝑐 ≠ 4 )
222 221 adantl ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑐 ≠ 4 )
223 222 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑐 = 4 )
224 223 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
225 3re ⊢ 3 ∈ ℝ
226 3lt4 ⊢ 3 < 4
227 225 226 ltneii ⊢ 3 ≠ 4
228 neeq1 ⊢ ( 𝑏 = 3 → ( 𝑏 ≠ 4 ↔ 3 ≠ 4 ) )
229 227 228 mpbiri ⊢ ( 𝑏 = 3 → 𝑏 ≠ 4 )
230 229 adantr ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → 𝑏 ≠ 4 )
231 230 neneqd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ¬ 𝑏 = 4 )
232 231 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
233 224 232 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
234 231 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
235 223 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
236 234 235 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
237 193 orcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
238 196 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
239 237 238 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
240 233 236 239 3jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
241 216 240 jca ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
242 194 197 241 jca31 ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
243 242 olcd ⊢ ( ( 𝑏 = 3 ∧ 𝑐 = 5 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
244 189 191 243 3jaodan ⊢ ( ( 𝑏 = 3 ∧ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
245 171 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑐 ≠ 3 )
246 245 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 3 )
247 246 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
248 141 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑐 ≠ 0 )
249 248 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 0 )
250 249 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
251 neeq1 ⊢ ( 𝑏 = 5 → ( 𝑏 ≠ 0 ↔ 5 ≠ 0 ) )
252 95 251 mpbiri ⊢ ( 𝑏 = 5 → 𝑏 ≠ 0 )
253 252 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑏 ≠ 0 )
254 253 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 0 )
255 254 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
256 249 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
257 255 256 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
258 74 necon2i ⊢ ( 𝑏 = 5 → 𝑏 ≠ 1 )
259 258 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑏 ≠ 1 )
260 259 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 1 )
261 260 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
262 neeq1 ⊢ ( 𝑏 = 5 → ( 𝑏 ≠ 2 ↔ 5 ≠ 2 ) )
263 103 262 mpbiri ⊢ ( 𝑏 = 5 → 𝑏 ≠ 2 )
264 263 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑏 ≠ 2 )
265 264 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 2 )
266 265 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
267 261 266 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
268 246 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
269 160 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑐 ≠ 2 )
270 269 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 2 )
271 270 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
272 268 271 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
273 257 267 272 3jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
274 3lt5 ⊢ 3 < 5
275 225 274 gtneii ⊢ 5 ≠ 3
276 neeq1 ⊢ ( 𝑏 = 5 → ( 𝑏 ≠ 3 ↔ 5 ≠ 3 ) )
277 275 276 mpbiri ⊢ ( 𝑏 = 5 → 𝑏 ≠ 3 )
278 277 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑏 ≠ 3 )
279 278 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑏 = 3 )
280 279 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
281 246 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
282 280 281 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
283 177 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑐 ≠ 5 )
284 283 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 5 )
285 284 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
286 167 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → 𝑐 ≠ 4 )
287 286 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ¬ 𝑐 = 4 )
288 287 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
289 285 288 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
290 254 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
291 249 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
292 290 291 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
293 282 289 292 3jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
294 273 293 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
295 247 250 294 jca31 ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
296 295 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 1 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
297 252 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑏 ≠ 0 )
298 297 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 0 )
299 298 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
300 23 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑐 ≠ 0 )
301 300 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 0 )
302 301 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
303 298 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
304 301 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
305 303 304 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
306 258 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑏 ≠ 1 )
307 306 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 1 )
308 307 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
309 42 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑐 ≠ 1 )
310 309 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 1 )
311 310 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
312 308 311 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
313 263 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑏 ≠ 2 )
314 313 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 2 )
315 314 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
316 277 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑏 ≠ 3 )
317 316 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 3 )
318 317 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
319 315 318 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
320 305 312 319 3jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
321 317 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
322 neeq1 ⊢ ( 𝑏 = 5 → ( 𝑏 ≠ 4 ↔ 5 ≠ 4 ) )
323 219 322 mpbiri ⊢ ( 𝑏 = 5 → 𝑏 ≠ 4 )
324 323 adantr ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑏 ≠ 4 )
325 324 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑏 = 4 )
326 325 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
327 321 326 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
328 325 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
329 neeq1 ⊢ ( 𝑐 = 3 → ( 𝑐 ≠ 4 ↔ 3 ≠ 4 ) )
330 227 329 mpbiri ⊢ ( 𝑐 = 3 → 𝑐 ≠ 4 )
331 330 adantl ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → 𝑐 ≠ 4 )
332 331 neneqd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ¬ 𝑐 = 4 )
333 332 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
334 328 333 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
335 298 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
336 301 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
337 335 336 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
338 327 334 337 3jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
339 320 338 jca ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
340 299 302 339 jca31 ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
341 340 olcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 3 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
342 eqtr3 ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 5 ) → 𝑏 = 𝑐 )
343 342 orcd ⊢ ( ( 𝑏 = 5 ∧ 𝑐 = 5 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
344 296 341 343 3jaodan ⊢ ( ( 𝑏 = 5 ∧ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
345 134 244 344 3jaoian ⊢ ( ( ( 𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5 ) ∧ ( 𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
346 8 12 345 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
347 346 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
348 1 2 3 usgrexmpl2nb1 ⊢ ( 𝐺 NeighbVtx 1 ) = { 0 , 2 }
349 348 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ↔ 𝑏 ∈ { 0 , 2 } )
350 6 elpr ⊢ ( 𝑏 ∈ { 0 , 2 } ↔ ( 𝑏 = 0 ∨ 𝑏 = 2 ) )
351 349 350 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ↔ ( 𝑏 = 0 ∨ 𝑏 = 2 ) )
352 348 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ↔ 𝑐 ∈ { 0 , 2 } )
353 10 elpr ⊢ ( 𝑐 ∈ { 0 , 2 } ↔ ( 𝑐 = 0 ∨ 𝑐 = 2 ) )
354 352 353 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ↔ ( 𝑐 = 0 ∨ 𝑐 = 2 ) )
355 eqtr3 ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 0 ) → 𝑏 = 𝑐 )
356 355 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 0 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
357 2ne0 ⊢ 2 ≠ 0
358 neeq1 ⊢ ( 𝑏 = 2 → ( 𝑏 ≠ 0 ↔ 2 ≠ 0 ) )
359 357 358 mpbiri ⊢ ( 𝑏 = 2 → 𝑏 ≠ 0 )
360 359 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑏 ≠ 0 )
361 360 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 0 )
362 361 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
363 153 necon2i ⊢ ( 𝑏 = 2 → 𝑏 ≠ 3 )
364 363 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑏 ≠ 3 )
365 364 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 3 )
366 365 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
367 361 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
368 49 necon2i ⊢ ( 𝑏 = 2 → 𝑏 ≠ 1 )
369 368 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑏 ≠ 1 )
370 369 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 1 )
371 370 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
372 367 371 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
373 370 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
374 141 necon2i ⊢ ( 𝑐 = 0 → 𝑐 ≠ 1 )
375 374 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑐 ≠ 1 )
376 375 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 1 )
377 376 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
378 373 377 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
379 23 necon2i ⊢ ( 𝑐 = 0 → 𝑐 ≠ 3 )
380 379 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑐 ≠ 3 )
381 380 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 3 )
382 381 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
383 365 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
384 382 383 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
385 372 378 384 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
386 365 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
387 381 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
388 386 387 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
389 97 necon2i ⊢ ( 𝑐 = 0 → 𝑐 ≠ 5 )
390 389 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑐 ≠ 5 )
391 390 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 5 )
392 391 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
393 4pos ⊢ 0 < 4
394 93 393 ltneii ⊢ 0 ≠ 4
395 neeq1 ⊢ ( 𝑐 = 0 → ( 𝑐 ≠ 4 ↔ 0 ≠ 4 ) )
396 394 395 mpbiri ⊢ ( 𝑐 = 0 → 𝑐 ≠ 4 )
397 396 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑐 ≠ 4 )
398 397 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 4 )
399 398 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
400 392 399 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
401 361 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
402 263 necon2i ⊢ ( 𝑏 = 2 → 𝑏 ≠ 5 )
403 402 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → 𝑏 ≠ 5 )
404 403 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 5 )
405 404 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
406 401 405 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
407 388 400 406 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
408 385 407 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
409 362 366 408 jca31 ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
410 409 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 0 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
411 34 necon2i ⊢ ( 𝑐 = 2 → 𝑐 ≠ 3 )
412 411 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑐 ≠ 3 )
413 412 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 3 )
414 413 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
415 neeq1 ⊢ ( 𝑐 = 2 → ( 𝑐 ≠ 0 ↔ 2 ≠ 0 ) )
416 357 415 mpbiri ⊢ ( 𝑐 = 2 → 𝑐 ≠ 0 )
417 416 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑐 ≠ 0 )
418 417 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 0 )
419 418 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
420 160 necon2i ⊢ ( 𝑐 = 2 → 𝑐 ≠ 1 )
421 420 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑐 ≠ 1 )
422 421 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 1 )
423 422 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
424 418 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
425 423 424 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
426 17 necon2i ⊢ ( 𝑏 = 0 → 𝑏 ≠ 1 )
427 426 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑏 ≠ 1 )
428 427 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 1 )
429 428 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
430 359 necon2i ⊢ ( 𝑏 = 0 → 𝑏 ≠ 2 )
431 430 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑏 ≠ 2 )
432 431 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 2 )
433 432 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
434 429 433 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
435 413 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
436 136 necon2i ⊢ ( 𝑏 = 0 → 𝑏 ≠ 3 )
437 436 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑏 ≠ 3 )
438 437 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 3 )
439 438 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
440 435 439 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
441 425 434 440 3jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
442 438 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
443 413 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
444 442 443 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
445 neeq1 ⊢ ( 𝑏 = 0 → ( 𝑏 ≠ 4 ↔ 0 ≠ 4 ) )
446 394 445 mpbiri ⊢ ( 𝑏 = 0 → 𝑏 ≠ 4 )
447 446 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑏 ≠ 4 )
448 447 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 4 )
449 448 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
450 252 necon2i ⊢ ( 𝑏 = 0 → 𝑏 ≠ 5 )
451 450 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑏 ≠ 5 )
452 451 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 5 )
453 452 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
454 449 453 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
455 105 necon2i ⊢ ( 𝑐 = 2 → 𝑐 ≠ 5 )
456 455 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → 𝑐 ≠ 5 )
457 456 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 5 )
458 457 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
459 418 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
460 458 459 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
461 444 454 460 3jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
462 441 461 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
463 414 419 462 jca31 ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
464 463 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 2 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
465 359 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑏 ≠ 0 )
466 465 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 0 )
467 466 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
468 416 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑐 ≠ 0 )
469 468 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 0 )
470 469 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
471 466 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
472 469 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
473 471 472 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
474 368 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑏 ≠ 1 )
475 474 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 1 )
476 475 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
477 420 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑐 ≠ 1 )
478 477 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 1 )
479 478 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
480 476 479 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
481 411 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑐 ≠ 3 )
482 481 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 3 )
483 482 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
484 363 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑏 ≠ 3 )
485 484 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 3 )
486 485 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
487 483 486 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
488 473 480 487 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
489 485 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
490 2lt4 ⊢ 2 < 4
491 30 490 ltneii ⊢ 2 ≠ 4
492 neeq1 ⊢ ( 𝑏 = 2 → ( 𝑏 ≠ 4 ↔ 2 ≠ 4 ) )
493 491 492 mpbiri ⊢ ( 𝑏 = 2 → 𝑏 ≠ 4 )
494 493 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑏 ≠ 4 )
495 494 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 4 )
496 495 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
497 489 496 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
498 495 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
499 402 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → 𝑏 ≠ 5 )
500 499 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 5 )
501 500 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
502 498 501 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
503 466 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
504 469 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
505 503 504 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
506 497 502 505 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
507 488 506 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
508 467 470 507 jca31 ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
509 508 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 2 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
510 356 410 464 509 ccase ⊢ ( ( ( 𝑏 = 0 ∨ 𝑏 = 2 ) ∧ ( 𝑐 = 0 ∨ 𝑐 = 2 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
511 351 354 510 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
512 511 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
513 1 2 3 usgrexmpl2nb2 ⊢ ( 𝐺 NeighbVtx 2 ) = { 1 , 3 }
514 513 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ↔ 𝑏 ∈ { 1 , 3 } )
515 6 elpr ⊢ ( 𝑏 ∈ { 1 , 3 } ↔ ( 𝑏 = 1 ∨ 𝑏 = 3 ) )
516 514 515 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ↔ ( 𝑏 = 1 ∨ 𝑏 = 3 ) )
517 513 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ↔ 𝑐 ∈ { 1 , 3 } )
518 10 elpr ⊢ ( 𝑐 ∈ { 1 , 3 } ↔ ( 𝑐 = 1 ∨ 𝑐 = 3 ) )
519 517 518 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ↔ ( 𝑐 = 1 ∨ 𝑐 = 3 ) )
520 14 189 85 191 ccase ⊢ ( ( ( 𝑏 = 1 ∨ 𝑏 = 3 ) ∧ ( 𝑐 = 1 ∨ 𝑐 = 3 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
521 516 519 520 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
522 521 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
523 c0ex ⊢ 0 ∈ V
524 1ex ⊢ 1 ∈ V
525 2ex ⊢ 2 ∈ V
526 oveq2 ⊢ ( 𝑎 = 0 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 0 ) )
527 526 raleqdv ⊢ ( 𝑎 = 0 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
528 526 527 raleqbidv ⊢ ( 𝑎 = 0 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
529 oveq2 ⊢ ( 𝑎 = 1 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 1 ) )
530 529 raleqdv ⊢ ( 𝑎 = 1 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
531 529 530 raleqbidv ⊢ ( 𝑎 = 1 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
532 oveq2 ⊢ ( 𝑎 = 2 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 2 ) )
533 532 raleqdv ⊢ ( 𝑎 = 2 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
534 532 533 raleqbidv ⊢ ( 𝑎 = 2 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
535 523 524 525 528 531 534 raltp ⊢ ( ∀ 𝑎 ∈ { 0 , 1 , 2 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 0 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 0 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ∧ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 1 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 1 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ∧ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 2 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 2 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
536 347 512 522 535 mpbir3an ⊢ ∀ 𝑎 ∈ { 0 , 1 , 2 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
537 1 2 3 usgrexmpl2nb3 ⊢ ( 𝐺 NeighbVtx 3 ) = { 0 , 2 , 4 }
538 537 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ↔ 𝑏 ∈ { 0 , 2 , 4 } )
539 6 eltp ⊢ ( 𝑏 ∈ { 0 , 2 , 4 } ↔ ( 𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4 ) )
540 538 539 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ↔ ( 𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4 ) )
541 537 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ↔ 𝑐 ∈ { 0 , 2 , 4 } )
542 10 eltp ⊢ ( 𝑐 ∈ { 0 , 2 , 4 } ↔ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) )
543 541 542 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ↔ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) )
544 330 necon2i ⊢ ( 𝑐 = 4 → 𝑐 ≠ 3 )
545 544 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑐 ≠ 3 )
546 545 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 3 )
547 546 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
548 436 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑏 ≠ 3 )
549 548 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 3 )
550 549 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
551 167 necon2i ⊢ ( 𝑐 = 4 → 𝑐 ≠ 1 )
552 551 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑐 ≠ 1 )
553 552 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 1 )
554 553 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
555 426 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑏 ≠ 1 )
556 555 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 1 )
557 556 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
558 554 557 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
559 556 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
560 430 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑏 ≠ 2 )
561 560 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 2 )
562 561 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
563 559 562 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
564 546 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
565 549 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
566 564 565 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
567 558 563 566 3jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
568 549 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
569 546 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
570 568 569 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
571 446 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑏 ≠ 4 )
572 571 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 4 )
573 572 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
574 450 adantr ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑏 ≠ 5 )
575 574 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 5 )
576 575 orcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
577 573 576 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
578 221 necon2i ⊢ ( 𝑐 = 4 → 𝑐 ≠ 5 )
579 578 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑐 ≠ 5 )
580 579 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 5 )
581 580 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
582 396 necon2i ⊢ ( 𝑐 = 4 → 𝑐 ≠ 0 )
583 582 adantl ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → 𝑐 ≠ 0 )
584 583 neneqd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 0 )
585 584 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
586 581 585 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
587 570 577 586 3jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
588 567 587 jca ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
589 547 550 588 jca31 ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
590 589 olcd ⊢ ( ( 𝑏 = 0 ∧ 𝑐 = 4 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
591 356 464 590 3jaodan ⊢ ( ( 𝑏 = 0 ∧ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
592 359 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑏 ≠ 0 )
593 592 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 0 )
594 593 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
595 582 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑐 ≠ 0 )
596 595 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 0 )
597 596 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
598 593 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
599 596 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
600 598 599 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
601 368 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑏 ≠ 1 )
602 601 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 1 )
603 602 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
604 551 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑐 ≠ 1 )
605 604 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 1 )
606 605 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
607 603 606 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
608 544 adantl ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑐 ≠ 3 )
609 608 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑐 = 3 )
610 609 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
611 363 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑏 ≠ 3 )
612 611 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 3 )
613 612 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
614 610 613 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
615 600 607 614 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
616 612 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
617 609 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
618 616 617 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
619 493 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑏 ≠ 4 )
620 619 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 4 )
621 620 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
622 402 adantr ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → 𝑏 ≠ 5 )
623 622 neneqd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ¬ 𝑏 = 5 )
624 623 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
625 621 624 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
626 593 orcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
627 596 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
628 626 627 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
629 618 625 628 3jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
630 615 629 jca ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
631 594 597 630 jca31 ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
632 631 olcd ⊢ ( ( 𝑏 = 2 ∧ 𝑐 = 4 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
633 410 509 632 3jaodan ⊢ ( ( 𝑏 = 2 ∧ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
634 446 necon2i ⊢ ( 𝑏 = 4 → 𝑏 ≠ 0 )
635 634 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑏 ≠ 0 )
636 635 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 0 )
637 636 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
638 229 necon2i ⊢ ( 𝑏 = 4 → 𝑏 ≠ 3 )
639 638 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑏 ≠ 3 )
640 639 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 3 )
641 640 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
642 636 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
643 65 necon2i ⊢ ( 𝑏 = 4 → 𝑏 ≠ 1 )
644 643 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑏 ≠ 1 )
645 644 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 1 )
646 645 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
647 642 646 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
648 416 necon2i ⊢ ( 𝑐 = 0 → 𝑐 ≠ 2 )
649 648 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑐 ≠ 2 )
650 649 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 2 )
651 650 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
652 374 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑐 ≠ 1 )
653 652 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 1 )
654 653 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
655 651 654 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
656 379 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑐 ≠ 3 )
657 656 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 3 )
658 657 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
659 640 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
660 658 659 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
661 647 655 660 3jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
662 640 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
663 657 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
664 662 663 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
665 389 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑐 ≠ 5 )
666 665 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 5 )
667 666 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
668 396 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑐 ≠ 4 )
669 668 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑐 = 4 )
670 669 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
671 667 670 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
672 636 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
673 323 necon2i ⊢ ( 𝑏 = 4 → 𝑏 ≠ 5 )
674 673 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → 𝑏 ≠ 5 )
675 674 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ¬ 𝑏 = 5 )
676 675 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
677 672 676 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
678 664 671 677 3jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
679 661 678 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
680 637 641 679 jca31 ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
681 680 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 0 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
682 634 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑏 ≠ 0 )
683 682 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 0 )
684 683 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
685 416 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑐 ≠ 0 )
686 685 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 0 )
687 686 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
688 683 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
689 686 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
690 688 689 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
691 643 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑏 ≠ 1 )
692 691 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 1 )
693 692 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
694 420 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑐 ≠ 1 )
695 694 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 1 )
696 695 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
697 693 696 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
698 493 necon2i ⊢ ( 𝑏 = 4 → 𝑏 ≠ 2 )
699 698 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑏 ≠ 2 )
700 699 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 2 )
701 700 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
702 638 adantr ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑏 ≠ 3 )
703 702 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑏 = 3 )
704 703 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
705 701 704 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
706 690 697 705 3jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
707 703 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
708 411 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑐 ≠ 3 )
709 708 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 3 )
710 709 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
711 707 710 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
712 455 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑐 ≠ 5 )
713 712 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 5 )
714 713 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
715 neeq1 ⊢ ( 𝑐 = 2 → ( 𝑐 ≠ 4 ↔ 2 ≠ 4 ) )
716 491 715 mpbiri ⊢ ( 𝑐 = 2 → 𝑐 ≠ 4 )
717 716 adantl ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → 𝑐 ≠ 4 )
718 717 neneqd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ¬ 𝑐 = 4 )
719 718 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
720 714 719 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
721 683 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
722 686 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
723 721 722 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
724 711 720 723 3jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
725 706 724 jca ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
726 684 687 725 jca31 ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
727 726 olcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 2 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
728 eqtr3 ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 4 ) → 𝑏 = 𝑐 )
729 728 orcd ⊢ ( ( 𝑏 = 4 ∧ 𝑐 = 4 ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
730 681 727 729 3jaodan ⊢ ( ( 𝑏 = 4 ∧ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
731 591 633 730 3jaoian ⊢ ( ( ( 𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4 ) ∧ ( 𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
732 540 543 731 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
733 732 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
734 1 2 3 usgrexmpl2nb4 ⊢ ( 𝐺 NeighbVtx 4 ) = { 3 , 5 }
735 734 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ↔ 𝑏 ∈ { 3 , 5 } )
736 6 elpr ⊢ ( 𝑏 ∈ { 3 , 5 } ↔ ( 𝑏 = 3 ∨ 𝑏 = 5 ) )
737 735 736 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ↔ ( 𝑏 = 3 ∨ 𝑏 = 5 ) )
738 734 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ↔ 𝑐 ∈ { 3 , 5 } )
739 10 elpr ⊢ ( 𝑐 ∈ { 3 , 5 } ↔ ( 𝑐 = 3 ∨ 𝑐 = 5 ) )
740 738 739 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ↔ ( 𝑐 = 3 ∨ 𝑐 = 5 ) )
741 191 341 243 343 ccase ⊢ ( ( ( 𝑏 = 3 ∨ 𝑏 = 5 ) ∧ ( 𝑐 = 3 ∨ 𝑐 = 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
742 737 740 741 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
743 742 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
744 1 2 3 usgrexmpl2nb5 ⊢ ( 𝐺 NeighbVtx 5 ) = { 0 , 4 }
745 744 eleq2i ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ↔ 𝑏 ∈ { 0 , 4 } )
746 6 elpr ⊢ ( 𝑏 ∈ { 0 , 4 } ↔ ( 𝑏 = 0 ∨ 𝑏 = 4 ) )
747 745 746 bitri ⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ↔ ( 𝑏 = 0 ∨ 𝑏 = 4 ) )
748 744 eleq2i ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ↔ 𝑐 ∈ { 0 , 4 } )
749 10 elpr ⊢ ( 𝑐 ∈ { 0 , 4 } ↔ ( 𝑐 = 0 ∨ 𝑐 = 4 ) )
750 748 749 bitri ⊢ ( 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ↔ ( 𝑐 = 0 ∨ 𝑐 = 4 ) )
751 356 681 590 729 ccase ⊢ ( ( ( 𝑏 = 0 ∨ 𝑏 = 4 ) ∧ ( 𝑐 = 0 ∨ 𝑐 = 4 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
752 747 750 751 syl2anb ⊢ ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ∧ 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ) → ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
753 752 rgen2 ⊢ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
754 3ex ⊢ 3 ∈ V
755 4nn0 ⊢ 4 ∈ ℕ0
756 755 elexi ⊢ 4 ∈ V
757 5nn0 ⊢ 5 ∈ ℕ0
758 757 elexi ⊢ 5 ∈ V
759 oveq2 ⊢ ( 𝑎 = 3 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 3 ) )
760 759 raleqdv ⊢ ( 𝑎 = 3 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
761 759 760 raleqbidv ⊢ ( 𝑎 = 3 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
762 oveq2 ⊢ ( 𝑎 = 4 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 4 ) )
763 762 raleqdv ⊢ ( 𝑎 = 4 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
764 762 763 raleqbidv ⊢ ( 𝑎 = 4 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
765 oveq2 ⊢ ( 𝑎 = 5 → ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 5 ) )
766 765 raleqdv ⊢ ( 𝑎 = 5 → ( ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
767 765 766 raleqbidv ⊢ ( 𝑎 = 5 → ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
768 754 756 758 761 764 767 raltp ⊢ ( ∀ 𝑎 ∈ { 3 , 4 , 5 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ( ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 3 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 3 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ∧ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 4 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 4 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ∧ ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 5 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 5 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
769 733 743 753 768 mpbir3an ⊢ ∀ 𝑎 ∈ { 3 , 4 , 5 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
770 ralunb ⊢ ( ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ( ∀ 𝑎 ∈ { 0 , 1 , 2 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ∧ ∀ 𝑎 ∈ { 3 , 4 , 5 } ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ) )
771 536 769 770 mpbir2an ⊢ ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
772 ianor ⊢ ( ¬ ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) ↔ ( ¬ 𝑏 ≠ 𝑐 ∨ ¬ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
773 nne ⊢ ( ¬ 𝑏 ≠ 𝑐 ↔ 𝑏 = 𝑐 )
774 ioran ⊢ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ) ↔ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∧ ¬ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ) )
775 ioran ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ↔ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∧ ¬ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) )
776 ianor ⊢ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 3 ) ↔ ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) )
777 ianor ⊢ ( ¬ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ↔ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) )
778 776 777 anbi12i ⊢ ( ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∧ ¬ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) )
779 775 778 bitri ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) )
780 ioran ⊢ ( ¬ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ↔ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∧ ¬ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) )
781 3ioran ⊢ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ↔ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∧ ¬ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∧ ¬ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) )
782 ioran ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ↔ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∧ ¬ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) )
783 ianor ⊢ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 1 ) ↔ ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) )
784 ianor ⊢ ( ¬ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ↔ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) )
785 783 784 anbi12i ⊢ ( ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∧ ¬ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
786 782 785 bitri ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) )
787 ioran ⊢ ( ¬ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ↔ ( ¬ ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∧ ¬ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) )
788 ianor ⊢ ( ¬ ( 𝑏 = 1 ∧ 𝑐 = 2 ) ↔ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) )
789 ianor ⊢ ( ¬ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ↔ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) )
790 788 789 anbi12i ⊢ ( ( ¬ ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∧ ¬ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ↔ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
791 787 790 bitri ⊢ ( ¬ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ↔ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) )
792 ioran ⊢ ( ¬ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ↔ ( ¬ ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∧ ¬ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) )
793 ianor ⊢ ( ¬ ( 𝑏 = 2 ∧ 𝑐 = 3 ) ↔ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) )
794 ianor ⊢ ( ¬ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ↔ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) )
795 793 794 anbi12i ⊢ ( ( ¬ ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∧ ¬ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ↔ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
796 792 795 bitri ⊢ ( ¬ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ↔ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) )
797 786 791 796 3anbi123i ⊢ ( ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∧ ¬ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∧ ¬ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
798 781 797 bitri ⊢ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) )
799 3ioran ⊢ ( ¬ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ↔ ( ¬ ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∧ ¬ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∧ ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) )
800 ioran ⊢ ( ¬ ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ↔ ( ¬ ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∧ ¬ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) )
801 ianor ⊢ ( ¬ ( 𝑏 = 3 ∧ 𝑐 = 4 ) ↔ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) )
802 ianor ⊢ ( ¬ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ↔ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) )
803 801 802 anbi12i ⊢ ( ( ¬ ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∧ ¬ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ↔ ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
804 800 803 bitri ⊢ ( ¬ ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ↔ ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) )
805 ioran ⊢ ( ¬ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ↔ ( ¬ ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∧ ¬ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) )
806 ianor ⊢ ( ¬ ( 𝑏 = 4 ∧ 𝑐 = 5 ) ↔ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) )
807 ianor ⊢ ( ¬ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ↔ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) )
808 806 807 anbi12i ⊢ ( ( ¬ ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∧ ¬ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ↔ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
809 805 808 bitri ⊢ ( ¬ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ↔ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) )
810 ioran ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ↔ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∧ ¬ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) )
811 ianor ⊢ ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 5 ) ↔ ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) )
812 ianor ⊢ ( ¬ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ↔ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) )
813 811 812 anbi12i ⊢ ( ( ¬ ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∧ ¬ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
814 810 813 bitri ⊢ ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ↔ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) )
815 804 809 814 3anbi123i ⊢ ( ( ¬ ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∧ ¬ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∧ ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ↔ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
816 799 815 bitri ⊢ ( ¬ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ↔ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) )
817 798 816 anbi12i ⊢ ( ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∧ ¬ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ↔ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
818 780 817 bitri ⊢ ( ¬ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ↔ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) )
819 779 818 anbi12i ⊢ ( ( ¬ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∧ ¬ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
820 774 819 bitri ⊢ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
821 6 10 523 524 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 0 , 1 } ↔ ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) )
822 6 10 524 525 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 1 , 2 } ↔ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) )
823 6 10 525 754 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 2 , 3 } ↔ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) )
824 821 822 823 3orbi123i ⊢ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ↔ ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) )
825 6 10 754 756 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 3 , 4 } ↔ ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) )
826 6 10 756 758 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 4 , 5 } ↔ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) )
827 6 10 523 758 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 0 , 5 } ↔ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) )
828 825 826 827 3orbi123i ⊢ ( ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ↔ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) )
829 824 828 orbi12i ⊢ ( ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) ↔ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) )
830 829 orbi2i ⊢ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) ) ↔ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( ( ( 𝑏 = 0 ∧ 𝑐 = 1 ) ∨ ( 𝑏 = 1 ∧ 𝑐 = 0 ) ) ∨ ( ( 𝑏 = 1 ∧ 𝑐 = 2 ) ∨ ( 𝑏 = 2 ∧ 𝑐 = 1 ) ) ∨ ( ( 𝑏 = 2 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 2 ) ) ) ∨ ( ( ( 𝑏 = 3 ∧ 𝑐 = 4 ) ∨ ( 𝑏 = 4 ∧ 𝑐 = 3 ) ) ∨ ( ( 𝑏 = 4 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 4 ) ) ∨ ( ( 𝑏 = 0 ∧ 𝑐 = 5 ) ∨ ( 𝑏 = 5 ∧ 𝑐 = 0 ) ) ) ) ) )
831 820 830 xchnxbir ⊢ ( ¬ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
832 elun ⊢ ( { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ↔ ( { 𝑏 , 𝑐 } ∈ { { 0 , 3 } } ∨ { 𝑏 , 𝑐 } ∈ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) )
833 prex ⊢ { 𝑏 , 𝑐 } ∈ V
834 833 elsn ⊢ ( { 𝑏 , 𝑐 } ∈ { { 0 , 3 } } ↔ { 𝑏 , 𝑐 } = { 0 , 3 } )
835 6 10 523 754 preq12b ⊢ ( { 𝑏 , 𝑐 } = { 0 , 3 } ↔ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) )
836 834 835 bitri ⊢ ( { 𝑏 , 𝑐 } ∈ { { 0 , 3 } } ↔ ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) )
837 elun ⊢ ( { 𝑏 , 𝑐 } ∈ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ↔ ( { 𝑏 , 𝑐 } ∈ { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∨ { 𝑏 , 𝑐 } ∈ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) )
838 833 eltp ⊢ ( { 𝑏 , 𝑐 } ∈ { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ↔ ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) )
839 833 eltp ⊢ ( { 𝑏 , 𝑐 } ∈ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ↔ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) )
840 838 839 orbi12i ⊢ ( ( { 𝑏 , 𝑐 } ∈ { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∨ { 𝑏 , 𝑐 } ∈ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ↔ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) )
841 837 840 bitri ⊢ ( { 𝑏 , 𝑐 } ∈ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ↔ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) )
842 836 841 orbi12i ⊢ ( ( { 𝑏 , 𝑐 } ∈ { { 0 , 3 } } ∨ { 𝑏 , 𝑐 } ∈ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ↔ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) ) )
843 832 842 bitri ⊢ ( { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ↔ ( ( ( 𝑏 = 0 ∧ 𝑐 = 3 ) ∨ ( 𝑏 = 3 ∧ 𝑐 = 0 ) ) ∨ ( ( { 𝑏 , 𝑐 } = { 0 , 1 } ∨ { 𝑏 , 𝑐 } = { 1 , 2 } ∨ { 𝑏 , 𝑐 } = { 2 , 3 } ) ∨ ( { 𝑏 , 𝑐 } = { 3 , 4 } ∨ { 𝑏 , 𝑐 } = { 4 , 5 } ∨ { 𝑏 , 𝑐 } = { 0 , 5 } ) ) ) )
844 831 843 xchnxbir ⊢ ( ¬ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ↔ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) )
845 773 844 orbi12i ⊢ ( ( ¬ 𝑏 ≠ 𝑐 ∨ ¬ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) ↔ ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) )
846 772 845 bitr2i ⊢ ( ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ¬ ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
847 846 3ralbii ⊢ ( ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ¬ ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
848 ralnex3 ⊢ ( ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ¬ ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) ↔ ¬ ∃ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∃ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∃ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
849 847 848 bitri ⊢ ( ∀ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∀ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∀ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 = 𝑐 ∨ ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1 ) ∧ ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0 ) ) ∧ ( ( ¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2 ) ∧ ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1 ) ) ∧ ( ( ¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3 ) ∧ ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2 ) ) ) ∧ ( ( ( ¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4 ) ∧ ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3 ) ) ∧ ( ( ¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4 ) ) ∧ ( ( ¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5 ) ∧ ( ¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0 ) ) ) ) ) ) ↔ ¬ ∃ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∃ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∃ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
850 771 849 mpbi ⊢ ¬ ∃ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∃ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∃ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) )
851 1 2 3 usgrexmpl2 ⊢ 𝐺 ∈ USGraph
852 1 2 3 usgrexmpl2vtx ⊢ ( Vtx ‘ 𝐺 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } )
853 852 eqcomi ⊢ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) = ( Vtx ‘ 𝐺 )
854 1 2 3 usgrexmpl2edg ⊢ ( Edg ‘ 𝐺 ) = ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) )
855 854 eqcomi ⊢ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) = ( Edg ‘ 𝐺 )
856 eqid ⊢ ( 𝐺 NeighbVtx 𝑎 ) = ( 𝐺 NeighbVtx 𝑎 )
857 853 855 856 usgrgrtrirex ⊢ ( 𝐺 ∈ USGraph → ( ∃ 𝑡 𝑡 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ∃ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∃ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∃ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) ) )
858 851 857 ax-mp ⊢ ( ∃ 𝑡 𝑡 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ∃ 𝑎 ∈ ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } ) ∃ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∃ 𝑐 ∈ ( 𝐺 NeighbVtx 𝑎 ) ( 𝑏 ≠ 𝑐 ∧ { 𝑏 , 𝑐 } ∈ ( { { 0 , 3 } } ∪ ( { { 0 , 1 } , { 1 , 2 } , { 2 , 3 } } ∪ { { 3 , 4 } , { 4 , 5 } , { 0 , 5 } } ) ) ) )
859 850 858 mtbir ⊢ ¬ ∃ 𝑡 𝑡 ∈ ( GrTriangles ‘ 𝐺 )