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Metamath Proof Explorer

Theorem findcard3

Description: Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013) Avoid ax-pow . (Revised by BTernaryTau, 7-Jan-2025)

Ref Expression
Hypotheses findcard3.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) )
findcard3.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) )
findcard3.3 ⊢ ( 𝑦 ∈ Fin → ( ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) → 𝜒 ) )
Assertion findcard3 ( 𝐴 ∈ Fin → 𝜏 )

Proof

Step Hyp Ref Expression
1 findcard3.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) )
2 findcard3.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) )
3 findcard3.3 ⊢ ( 𝑦 ∈ Fin → ( ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) → 𝜒 ) )
4 isfi ⊢ ( 𝐴 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 )
5 nnon ⊢ ( 𝑤 ∈ ω → 𝑤 ∈ On )
6 eleq1w ⊢ ( 𝑤 = 𝑧 → ( 𝑤 ∈ ω ↔ 𝑧 ∈ ω ) )
7 breq2 ⊢ ( 𝑤 = 𝑧 → ( 𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑧 ) )
8 7 imbi1d ⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
9 8 albidv ⊢ ( 𝑤 = 𝑧 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
10 6 9 imbi12d ⊢ ( 𝑤 = 𝑧 → ( ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ↔ ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) )
11 rspe ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ∃ 𝑤 ∈ ω 𝑦 ≈ 𝑤 )
12 isfi ⊢ ( 𝑦 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝑦 ≈ 𝑤 )
13 11 12 sylibr ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → 𝑦 ∈ Fin )
14 19.21v ⊢ ( ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
15 14 ralbii ⊢ ( ∀ 𝑧 ∈ 𝑤 ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
16 ralcom4 ⊢ ( ∀ 𝑧 ∈ 𝑤 ∀ 𝑥 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
17 15 16 bitr3i ⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
18 pssss ⊢ ( 𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦 )
19 ssfi ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊆ 𝑦 ) → 𝑥 ∈ Fin )
20 isfi ⊢ ( 𝑥 ∈ Fin ↔ ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 )
21 19 20 sylib ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊆ 𝑦 ) → ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 )
22 13 18 21 syl2an ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 )
23 simprl ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ∈ ω )
24 nnfi ⊢ ( 𝑧 ∈ ω → 𝑧 ∈ Fin )
25 ensymfib ⊢ ( 𝑧 ∈ Fin → ( 𝑧 ≈ 𝑥 ↔ 𝑥 ≈ 𝑧 ) )
26 24 25 syl ⊢ ( 𝑧 ∈ ω → ( 𝑧 ≈ 𝑥 ↔ 𝑥 ≈ 𝑧 ) )
27 26 biimpar ⊢ ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑧 ≈ 𝑥 )
28 27 adantl ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ≈ 𝑥 )
29 simplll ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑤 ∈ ω )
30 php3 ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ⊊ 𝑦 ) → 𝑥 ≺ 𝑦 )
31 13 30 sylan ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → 𝑥 ≺ 𝑦 )
32 31 adantr ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑥 ≺ 𝑦 )
33 simpllr ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑦 ≈ 𝑤 )
34 endom ⊢ ( 𝑦 ≈ 𝑤 → 𝑦 ≼ 𝑤 )
35 nnfi ⊢ ( 𝑤 ∈ ω → 𝑤 ∈ Fin )
36 domfi ⊢ ( ( 𝑤 ∈ Fin ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin )
37 35 36 sylan ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin )
38 37 3adant2 ⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑦 ∈ Fin )
39 sdomdom ⊢ ( 𝑥 ≺ 𝑦 → 𝑥 ≼ 𝑦 )
40 domfi ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≼ 𝑦 ) → 𝑥 ∈ Fin )
41 39 40 sylan2 ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≺ 𝑦 ) → 𝑥 ∈ Fin )
42 41 3adant3 ⊢ ( ( 𝑦 ∈ Fin ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ∈ Fin )
43 38 42 syld3an1 ⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ∈ Fin )
44 sdomdomtrfi ⊢ ( ( 𝑥 ∈ Fin ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ≺ 𝑤 )
45 43 44 syld3an1 ⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≼ 𝑤 ) → 𝑥 ≺ 𝑤 )
46 34 45 syl3an3 ⊢ ( ( 𝑤 ∈ ω ∧ 𝑥 ≺ 𝑦 ∧ 𝑦 ≈ 𝑤 ) → 𝑥 ≺ 𝑤 )
47 29 32 33 46 syl3anc ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑥 ≺ 𝑤 )
48 endom ⊢ ( 𝑧 ≈ 𝑥 → 𝑧 ≼ 𝑥 )
49 domsdomtrfi ⊢ ( ( 𝑧 ∈ Fin ∧ 𝑧 ≼ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 )
50 24 49 syl3an1 ⊢ ( ( 𝑧 ∈ ω ∧ 𝑧 ≼ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 )
51 48 50 syl3an2 ⊢ ( ( 𝑧 ∈ ω ∧ 𝑧 ≈ 𝑥 ∧ 𝑥 ≺ 𝑤 ) → 𝑧 ≺ 𝑤 )
52 23 28 47 51 syl3anc ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ≺ 𝑤 )
53 nnsdomo ⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) )
54 nnord ⊢ ( 𝑧 ∈ ω → Ord 𝑧 )
55 nnord ⊢ ( 𝑤 ∈ ω → Ord 𝑤 )
56 ordelpss ⊢ ( ( Ord 𝑧 ∧ Ord 𝑤 ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) )
57 54 55 56 syl2an ⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ⊊ 𝑤 ) )
58 53 57 bitr4d ⊢ ( ( 𝑧 ∈ ω ∧ 𝑤 ∈ ω ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ∈ 𝑤 ) )
59 23 29 58 syl2anc ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → ( 𝑧 ≺ 𝑤 ↔ 𝑧 ∈ 𝑤 ) )
60 52 59 mpbid ⊢ ( ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) ∧ ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) ) → 𝑧 ∈ 𝑤 )
61 60 ex ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑧 ∈ 𝑤 ) )
62 simpr ⊢ ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → 𝑥 ≈ 𝑧 )
63 61 62 jca2 ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ( 𝑧 ∈ ω ∧ 𝑥 ≈ 𝑧 ) → ( 𝑧 ∈ 𝑤 ∧ 𝑥 ≈ 𝑧 ) ) )
64 63 reximdv2 ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∃ 𝑧 ∈ ω 𝑥 ≈ 𝑧 → ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 ) )
65 22 64 mpd ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 )
66 r19.29 ⊢ ( ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) )
67 66 expcom ⊢ ( ∃ 𝑧 ∈ 𝑤 𝑥 ≈ 𝑧 → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) ) )
68 65 67 syl ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) ) )
69 ordom ⊢ Ord ω
70 ordelss ⊢ ( ( Ord ω ∧ 𝑤 ∈ ω ) → 𝑤 ⊆ ω )
71 69 70 mpan ⊢ ( 𝑤 ∈ ω → 𝑤 ⊆ ω )
72 71 ad2antrr ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → 𝑤 ⊆ ω )
73 72 sseld ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( 𝑧 ∈ 𝑤 → 𝑧 ∈ ω ) )
74 pm2.27 ⊢ ( 𝑧 ∈ ω → ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) )
75 74 impd ⊢ ( 𝑧 ∈ ω → ( ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) )
76 73 75 syl6 ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( 𝑧 ∈ 𝑤 → ( ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) ) )
77 76 rexlimdv ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∃ 𝑧 ∈ 𝑤 ( ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ∧ 𝑥 ≈ 𝑧 ) → 𝜑 ) )
78 68 77 syld ⊢ ( ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) ∧ 𝑥 ⊊ 𝑦 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜑 ) )
79 78 ex ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( 𝑥 ⊊ 𝑦 → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜑 ) ) )
80 79 com23 ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) )
81 80 alimdv ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑥 ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) )
82 17 81 biimtrid ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ⊊ 𝑦 → 𝜑 ) ) )
83 13 82 3 sylsyld ⊢ ( ( 𝑤 ∈ ω ∧ 𝑦 ≈ 𝑤 ) → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → 𝜒 ) )
84 83 impancom ⊢ ( ( 𝑤 ∈ ω ∧ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) → ( 𝑦 ≈ 𝑤 → 𝜒 ) )
85 84 alrimiv ⊢ ( ( 𝑤 ∈ ω ∧ ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) ) → ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) )
86 85 expcom ⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) ) )
87 breq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≈ 𝑤 ↔ 𝑦 ≈ 𝑤 ) )
88 87 1 imbi12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑦 ≈ 𝑤 → 𝜒 ) ) )
89 88 cbvalvw ⊢ ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ≈ 𝑤 → 𝜒 ) )
90 86 89 imbitrrdi ⊢ ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) )
91 90 a1i ⊢ ( 𝑤 ∈ On → ( ∀ 𝑧 ∈ 𝑤 ( 𝑧 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑧 → 𝜑 ) ) → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) ) )
92 10 91 tfis2 ⊢ ( 𝑤 ∈ On → ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) )
93 5 92 mpcom ⊢ ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) )
94 93 rgen ⊢ ∀ 𝑤 ∈ ω ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 )
95 r19.29 ⊢ ( ( ∀ 𝑤 ∈ ω ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 ) → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) )
96 94 95 mpan ⊢ ( ∃ 𝑤 ∈ ω 𝐴 ≈ 𝑤 → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) )
97 4 96 sylbi ⊢ ( 𝐴 ∈ Fin → ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) )
98 breq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≈ 𝑤 ↔ 𝐴 ≈ 𝑤 ) )
99 98 2 imbi12d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝐴 ≈ 𝑤 → 𝜏 ) ) )
100 99 spcgv ⊢ ( 𝐴 ∈ Fin → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) → ( 𝐴 ≈ 𝑤 → 𝜏 ) ) )
101 100 impd ⊢ ( 𝐴 ∈ Fin → ( ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) → 𝜏 ) )
102 101 rexlimdvw ⊢ ( 𝐴 ∈ Fin → ( ∃ 𝑤 ∈ ω ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ∧ 𝐴 ≈ 𝑤 ) → 𝜏 ) )
103 97 102 mpd ⊢ ( 𝐴 ∈ Fin → 𝜏 )