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Metamath Proof Explorer

Theorem axacndlem4

Description: Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 8-Jan-2002) (Proof shortened by Mario Carneiro, 10-Dec-2016)

Ref Expression
Assertion axacndlem4 ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) )

Proof

Step Hyp Ref Expression
1 zfac ⊢ ∃ 𝑣 ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) )
2 nfnae ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑧
3 nfnae ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
4 nfnae ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑤
5 2 3 4 nf3an ⊢ Ⅎ 𝑥 ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 )
6 nfnae ⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑧
7 nfnae ⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦
8 nfnae ⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑤
9 6 7 8 nf3an ⊢ Ⅎ 𝑦 ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 )
10 nfnae ⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑧
11 nfnae ⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦
12 nfnae ⊢ Ⅎ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑤
13 10 11 12 nf3an ⊢ Ⅎ 𝑧 ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 )
14 nfcvf ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 )
15 14 3ad2ant2 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑦 )
16 nfcvf ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑥 𝑧 )
17 16 3ad2ant1 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑧 )
18 15 17 nfeld ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑦 ∈ 𝑧 )
19 nfcvf ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑤 → Ⅎ 𝑥 𝑤 )
20 19 3ad2ant3 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑤 )
21 17 20 nfeld ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑧 ∈ 𝑤 )
22 18 21 nfand ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) )
23 nfnae ⊢ Ⅎ 𝑤 ¬ ∀ 𝑥 𝑥 = 𝑧
24 nfnae ⊢ Ⅎ 𝑤 ¬ ∀ 𝑥 𝑥 = 𝑦
25 nfnae ⊢ Ⅎ 𝑤 ¬ ∀ 𝑥 𝑥 = 𝑤
26 23 24 25 nf3an ⊢ Ⅎ 𝑤 ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 )
27 15 20 nfeld ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑦 ∈ 𝑤 )
28 nfcvd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑣 )
29 20 28 nfeld ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑤 ∈ 𝑣 )
30 27 29 nfand ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) )
31 22 30 nfand ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) )
32 26 31 nfexd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) )
33 15 20 nfeqd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 𝑦 = 𝑤 )
34 32 33 nfbid ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) )
35 9 34 nfald ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) )
36 26 35 nfexd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) )
37 22 36 nfimd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) )
38 13 37 nfald ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) )
39 9 38 nfald ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) )
40 nfcvd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑦 𝑣 )
41 nfcvf2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑦 𝑥 )
42 41 3ad2ant2 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑦 𝑥 )
43 40 42 nfeqd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑦 𝑣 = 𝑥 )
44 9 43 nfan1 ⊢ Ⅎ 𝑦 ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 )
45 nfcvd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑧 𝑣 )
46 nfcvf2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → Ⅎ 𝑧 𝑥 )
47 46 3ad2ant1 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑧 𝑥 )
48 45 47 nfeqd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑧 𝑣 = 𝑥 )
49 13 48 nfan1 ⊢ Ⅎ 𝑧 ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 )
50 22 nf5rd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ) )
51 50 adantr ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ) )
52 sp ⊢ ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) )
53 51 52 impbid1 ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ↔ ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ) )
54 nfcvd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑤 𝑣 )
55 nfcvf2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑤 → Ⅎ 𝑤 𝑥 )
56 55 3ad2ant3 ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑤 𝑥 )
57 54 56 nfeqd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → Ⅎ 𝑤 𝑣 = 𝑥 )
58 26 57 nfan1 ⊢ Ⅎ 𝑤 ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 )
59 simpr ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → 𝑣 = 𝑥 )
60 59 eleq2d ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( 𝑤 ∈ 𝑣 ↔ 𝑤 ∈ 𝑥 ) )
61 60 anbi2d ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ↔ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
62 61 anbi2d ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ) )
63 58 62 exbid ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ) )
64 63 bibi1d ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ↔ ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
65 44 64 albid ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ↔ ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
66 58 65 exbid ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ↔ ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
67 53 66 imbi12d ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) ↔ ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) )
68 49 67 albid ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) )
69 44 68 albid ⊢ ( ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) ∧ 𝑣 = 𝑥 ) → ( ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) )
70 69 ex ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → ( 𝑣 = 𝑥 → ( ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) ) )
71 5 39 70 cbvexd ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → ( ∃ 𝑣 ∀ 𝑦 ∀ 𝑧 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑣 ) ) ↔ 𝑦 = 𝑤 ) ) ↔ ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) )
72 1 71 mpbii ⊢ ( ( ¬ ∀ 𝑥 𝑥 = 𝑧 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ∧ ¬ ∀ 𝑥 𝑥 = 𝑤 ) → ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
73 72 3exp ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑧 → ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝑥 = 𝑤 → ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) ) ) )
74 axacndlem2 ⊢ ( ∀ 𝑥 𝑥 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
75 axacndlem1 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
76 nfae ⊢ Ⅎ 𝑦 ∀ 𝑥 𝑥 = 𝑤
77 nfae ⊢ Ⅎ 𝑧 ∀ 𝑥 𝑥 = 𝑤
78 simpr ⊢ ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → 𝑧 ∈ 𝑤 )
79 78 alimi ⊢ ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∀ 𝑥 𝑧 ∈ 𝑤 )
80 nd2 ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ¬ ∀ 𝑥 𝑧 ∈ 𝑤 )
81 80 pm2.21d ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ( ∀ 𝑥 𝑧 ∈ 𝑤 → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
82 79 81 syl5 ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
83 77 82 alrimi ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
84 76 83 alrimi ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
85 84 19.8ad ⊢ ( ∀ 𝑥 𝑥 = 𝑤 → ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) ) )
86 73 74 75 85 pm2.61iii ⊢ ∃ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ∀ 𝑥 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) → ∃ 𝑤 ∀ 𝑦 ( ∃ 𝑤 ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤 ) ∧ ( 𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) ↔ 𝑦 = 𝑤 ) )