Metamath Proof Explorer

Theorem copsexgw

Description: Version of copsexg with a disjoint variable condition, which does not require ax-13 . (Contributed by GG, 26-Jan-2024) Shorten proof and remove dependency on ax-10 . (Revised by Eric Schmidt, 2-May-2026)

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 anass ⊢ ( ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
13 12 exbii ⊢ ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
14 19.42v ⊢ ( ∃ 𝑦 ( 𝑧 = 𝑥 ∧ ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ↔ ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
15 13 14 bitri ⊢ ( ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
16 15 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
17 euequ ⊢ ∃! 𝑥 𝑥 = 𝑧
18 equcom ⊢ ( 𝑥 = 𝑧 ↔ 𝑧 = 𝑥 )
19 18 eubii ⊢ ( ∃! 𝑥 𝑥 = 𝑧 ↔ ∃! 𝑥 𝑧 = 𝑥 )
20 17 19 mpbi ⊢ ∃! 𝑥 𝑧 = 𝑥
21 eupick ⊢ ( ( ∃! 𝑥 𝑧 = 𝑥 ∧ ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) ) → ( 𝑧 = 𝑥 → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
22 20 21 mpan ⊢ ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑧 = 𝑥 → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
23 22 com12 ⊢ ( 𝑧 = 𝑥 → ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) )
24 euequ ⊢ ∃! 𝑦 𝑦 = 𝑤
25 equcom ⊢ ( 𝑦 = 𝑤 ↔ 𝑤 = 𝑦 )
26 25 eubii ⊢ ( ∃! 𝑦 𝑦 = 𝑤 ↔ ∃! 𝑦 𝑤 = 𝑦 )
27 24 26 mpbi ⊢ ∃! 𝑦 𝑤 = 𝑦
28 eupick ⊢ ( ( ∃! 𝑦 𝑤 = 𝑦 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → ( 𝑤 = 𝑦 → 𝜑 ) )
29 27 28 mpan ⊢ ( ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) → ( 𝑤 = 𝑦 → 𝜑 ) )
30 29 com12 ⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) → 𝜑 ) )
31 23 30 sylan9 ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ( 𝑧 = 𝑥 ∧ ∃ 𝑦 ( 𝑤 = 𝑦 ∧ 𝜑 ) ) → 𝜑 ) )
32 16 31 biimtrid ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ∧ 𝜑 ) → 𝜑 ) )
33 11 32 biimtrid ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝜑 ) )
34 9 33 sylbi ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) → 𝜑 ) )
35 6 34 impbid ⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
36 eqeq1 ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) )
37 36 anbi1d ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
38 37 2exbidv ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
39 38 bibi2d ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ↔ ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
40 36 39 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 ) ) ) ) )
41 35 40 mpbiri ⊢ ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
42 41 adantr ⊢ ( ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
43 42 exlimivv ⊢ ( ∃ 𝑧 ∃ 𝑤 ( 𝐴 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ) → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
44 3 43 sylbi ⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) ) )
45 44 pm2.43i ⊢ ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( 𝐴 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )