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Metamath Proof Explorer

Theorem dalem25

Description: Lemma for dath . Show that the dummy center of perspectivity c is different from auxiliary atom G . (Contributed by NM, 3-Aug-2012)

Ref Expression
Hypotheses dalem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
dalem.l ⊢ ≤ = ( le ‘ 𝐾 )
dalem.j ⊢ ∨ = ( join ‘ 𝐾 )
dalem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dalem.ps ⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
dalem23.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem23.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem23.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem23.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem23.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
Assertion dalem25 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≠ 𝐺 )

Proof

Step Hyp Ref Expression
1 dalem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
2 dalem.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dalem.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dalem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem.ps ⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
6 dalem23.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem23.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem23.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem23.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem23.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
11 1 2 3 4 dalemcnes ⊢ ( 𝜑 → 𝐶 ≠ 𝑆 )
12 11 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ≠ 𝑆 )
13 5 dalemclccjdd ⊢ ( 𝜓 → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
14 13 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
15 14 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
16 simpr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝑐 = 𝐺 )
17 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
18 17 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat )
19 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
20 19 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
21 5 dalemccea ⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
22 21 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 )
23 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
24 23 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ∈ 𝐴 )
25 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
26 25 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
27 20 22 24 26 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
28 5 dalemddea ⊢ ( 𝜓 → 𝑑 ∈ 𝐴 )
29 28 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 )
30 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
31 30 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ∈ 𝐴 )
32 25 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
33 20 29 31 32 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
34 25 2 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ≤ ( 𝑑 ∨ 𝑆 ) )
35 18 27 33 34 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ≤ ( 𝑑 ∨ 𝑆 ) )
36 10 35 eqbrtrid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≤ ( 𝑑 ∨ 𝑆 ) )
37 3 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) )
38 20 29 31 37 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) )
39 36 38 breqtrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≤ ( 𝑆 ∨ 𝑑 ) )
40 39 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝐺 ≤ ( 𝑆 ∨ 𝑑 ) )
41 16 40 eqbrtrd ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝑐 ≤ ( 𝑆 ∨ 𝑑 ) )
42 2 3 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) → 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) )
43 20 31 29 42 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) )
44 43 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) )
45 5 4 dalemcceb ⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) )
46 45 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ ( Base ‘ 𝐾 ) )
47 25 4 atbase ⊢ ( 𝑑 ∈ 𝐴 → 𝑑 ∈ ( Base ‘ 𝐾 ) )
48 28 47 syl ⊢ ( 𝜓 → 𝑑 ∈ ( Base ‘ 𝐾 ) )
49 48 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ ( Base ‘ 𝐾 ) )
50 25 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) → ( 𝑆 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) )
51 20 31 29 50 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑆 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) )
52 25 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑐 ∈ ( Base ‘ 𝐾 ) ∧ 𝑑 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑐 ≤ ( 𝑆 ∨ 𝑑 ) ∧ 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) ) ↔ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) )
53 18 46 49 51 52 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ≤ ( 𝑆 ∨ 𝑑 ) ∧ 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) ) ↔ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) )
54 53 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( ( 𝑐 ≤ ( 𝑆 ∨ 𝑑 ) ∧ 𝑑 ≤ ( 𝑆 ∨ 𝑑 ) ) ↔ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) )
55 41 44 54 mpbi2and ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) )
56 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
57 56 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ∈ ( Base ‘ 𝐾 ) )
58 25 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) )
59 20 22 29 58 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) )
60 25 2 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑐 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ∧ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) → 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) ) )
61 18 57 59 51 60 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ∧ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) → 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) ) )
62 61 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( ( 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ∧ ( 𝑐 ∨ 𝑑 ) ≤ ( 𝑆 ∨ 𝑑 ) ) → 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) ) )
63 15 55 62 mp2and ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) )
64 1 7 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
65 64 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) )
66 25 2 6 latmlem1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑑 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) → ( 𝐶 ∧ 𝑌 ) ≤ ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) ) )
67 18 57 51 65 66 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) → ( 𝐶 ∧ 𝑌 ) ≤ ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) ) )
68 67 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( 𝐶 ≤ ( 𝑆 ∨ 𝑑 ) → ( 𝐶 ∧ 𝑌 ) ≤ ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) ) )
69 63 68 mpd ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( 𝐶 ∧ 𝑌 ) ≤ ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) )
70 1 2 3 4 7 8 9 dalem17 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ 𝑌 )
71 70 3adant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ≤ 𝑌 )
72 25 2 6 latleeqm1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐶 ≤ 𝑌 ↔ ( 𝐶 ∧ 𝑌 ) = 𝐶 ) )
73 18 57 65 72 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐶 ≤ 𝑌 ↔ ( 𝐶 ∧ 𝑌 ) = 𝐶 ) )
74 71 73 mpbid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐶 ∧ 𝑌 ) = 𝐶 )
75 74 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( 𝐶 ∧ 𝑌 ) = 𝐶 )
76 1 2 3 4 9 dalemsly ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 𝑌 )
77 76 3adant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ≤ 𝑌 )
78 5 dalem-ddly ⊢ ( 𝜓 → ¬ 𝑑 ≤ 𝑌 )
79 78 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑌 )
80 25 2 3 6 4 2atjm ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌 ) ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 )
81 20 31 29 65 77 79 80 syl132anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 )
82 81 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 )
83 69 75 82 3brtr3d ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝐶 ≤ 𝑆 )
84 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
85 19 84 syl ⊢ ( 𝜑 → 𝐾 ∈ AtLat )
86 1 2 3 4 7 8 dalemcea ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
87 2 4 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆 ) )
88 85 86 30 87 syl3anc ⊢ ( 𝜑 → ( 𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆 ) )
89 88 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆 ) )
90 89 adantr ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → ( 𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆 ) )
91 83 90 mpbid ⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) ∧ 𝑐 = 𝐺 ) → 𝐶 = 𝑆 )
92 91 ex ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 = 𝐺 → 𝐶 = 𝑆 ) )
93 92 necon3d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐶 ≠ 𝑆 → 𝑐 ≠ 𝐺 ) )
94 12 93 mpd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≠ 𝐺 )