Metamath Proof Explorer

Theorem copsexgw

Description: Version of copsexg with a disjoint variable condition, which does not require ax-13 . (Contributed by Gino Giotto, 26-Jan-2024)

Ref Expression
Assertion copsexgw ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 vex ⊢ 𝑥 ∈ V
2 vex ⊢ 𝑦 ∈ V
3 1 2 eqvinop ⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ↔ ∃ 𝑧 ∃ 𝑤 ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) )
4 19.8a ⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
5 4 19.8ad ⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
6 5 ex ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 → ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
7 vex ⊢ 𝑧 ∈ V
8 vex ⊢ 𝑤 ∈ V
9 7 8 opth ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) )
10 9 anbi1i ⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) )
11 10 2exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) )
12 nfe1 ⊢ Ⅎ 𝑥 ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) )
13 19.8a ⊢ ( ( 𝑤 = 𝑦 ∧ 𝜑 ) → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) )
14 13 anim2i ⊢ ( ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
15 14 anassrs ⊢ ( ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
16 15 eximi ⊢ ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
17 biidd ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ↔ ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
18 17 drex1v ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ↔ ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
19 16 18 syl5ib ⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
20 anass ⊢ ( ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
21 20 exbii ⊢ ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
22 19.40 ⊢ ( ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( ∃ 𝑦 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
23 nfvd ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → Ⅎ 𝑦 𝑧 = 𝑥 )
24 23 19.9d ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 𝑧 = 𝑥 → 𝑧 = 𝑥 ) )
25 24 anim1d ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ( ∃ 𝑦 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
26 22 25 syl5 ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
27 21 26 syl5bi ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
28 19.8a ⊢ ( ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
29 27 28 syl6 ⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑥 → ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) )
30 19 29 pm2.61i ⊢ ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
31 12 30 exlimi ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
32 euequ ⊢ ∃! 𝑥 𝑥 = 𝑧
33 equcom ⊢ ( 𝑥 = 𝑧 ↔ 𝑧 = 𝑥 )
34 33 eubii ⊢ ( ∃! 𝑥 𝑥 = 𝑧 ↔ ∃! 𝑥 𝑧 = 𝑥 )
35 32 34 mpbi ⊢ ∃! 𝑥 𝑧 = 𝑥
36 eupick ⊢ ( ( ∃! 𝑥 𝑧 = 𝑥 ∧ ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) → ( 𝑧 = 𝑥 → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
37 35 36 mpan ⊢ ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑧 = 𝑥 → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
38 37 com12 ⊢ ( 𝑧 = 𝑥 → ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
39 euequ ⊢ ∃! 𝑦 𝑦 = 𝑤
40 equcom ⊢ ( 𝑦 = 𝑤 ↔ 𝑤 = 𝑦 )
41 40 eubii ⊢ ( ∃! 𝑦 𝑦 = 𝑤 ↔ ∃! 𝑦 𝑤 = 𝑦 )
42 39 41 mpbi ⊢ ∃! 𝑦 𝑤 = 𝑦
43 eupick ⊢ ( ( ∃! 𝑦 𝑤 = 𝑦 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑤 = 𝑦 → 𝜑 ) )
44 42 43 mpan ⊢ ( ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) → ( 𝑤 = 𝑦 → 𝜑 ) )
45 44 com12 ⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) → 𝜑 ) )
46 38 45 sylan9 ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → 𝜑 ) )
47 31 46 syl5 ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → 𝜑 ) )
48 11 47 syl5bi ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝜑 ) )
49 9 48 sylbi ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝜑 ) )
50 6 49 impbid ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
51 eqeq1 ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) )
52 51 anbi1d ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
53 52 2exbidv ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
54 53 bibi2d ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ↔ ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
55 51 54 imbi12d ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) ↔ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) ) )
56 50 55 mpbiri ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
57 56 adantr ⊢ ( ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
58 57 exlimivv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
59 3 58 sylbi ⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
60 59 pm2.43i ⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )