Metamath Proof Explorer

Theorem copsex2d

Description: Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023)

Ref Expression
Hypotheses copsex2d.xph ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
copsex2d.yph ⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
copsex2d.xch ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
copsex2d.ych ⊢ ( 𝜑 → Ⅎ 𝑦 𝜒 )
copsex2d.exa ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
copsex2d.exb ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
copsex2d.is ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion copsex2d ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )

Proof

Step Hyp Ref Expression
1 copsex2d.xph ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2 copsex2d.yph ⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3 copsex2d.xch ⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
4 copsex2d.ych ⊢ ( 𝜑 → Ⅎ 𝑦 𝜒 )
5 copsex2d.exa ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
6 copsex2d.exb ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
7 copsex2d.is ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
8 elisset ⊢ ( 𝐴 ∈ 𝑈 → ∃ 𝑥 𝑥 = 𝐴 )
9 5 8 syl ⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝐴 )
10 elisset ⊢ ( 𝐵 ∈ 𝑉 → ∃ 𝑦 𝑦 = 𝐵 )
11 6 10 syl ⊢ ( 𝜑 → ∃ 𝑦 𝑦 = 𝐵 )
12 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
13 nfe1 ⊢ Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 )
14 13 a1i ⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
15 14 3 nfbid ⊢ ( 𝜑 → Ⅎ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
16 15 19.9d ⊢ ( 𝜑 → ( ∃ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
17 nfe1 ⊢ Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 )
18 17 a1i ⊢ ( 𝜑 → Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
19 1 18 bj-nfexd ⊢ ( 𝜑 → Ⅎ 𝑦 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) )
20 19 4 nfbid ⊢ ( 𝜑 → Ⅎ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
21 20 19.9d ⊢ ( 𝜑 → ( ∃ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
22 opeq12 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
23 copsexgw ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜓 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ) )
24 23 bicomd ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
25 24 eqcoms ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
26 22 25 syl ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
27 26 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜓 ) )
28 27 7 bitrd ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
29 28 ex ⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
30 2 21 29 bj-exlimd ⊢ ( 𝜑 → ( ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
31 1 16 30 bj-exlimd ⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
32 12 31 syl5bir ⊢ ( 𝜑 → ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) ) )
33 9 11 32 mp2and ⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )