Metamath Proof Explorer

Theorem copsex2t

Description: Closed theorem form of copsex2g . (Contributed by NM, 17-Feb-2013)

Ref Expression
Assertion copsex2t ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 nfa1 ⊢ Ⅎ 𝑥 ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2 nfe1 ⊢ Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
3 nfv ⊢ Ⅎ 𝑥 𝜓
4 2 3 nfbi ⊢ Ⅎ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
5 nfa2 ⊢ Ⅎ 𝑦 ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
6 nfe1 ⊢ Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
7 6 nfex ⊢ Ⅎ 𝑦 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
8 nfv ⊢ Ⅎ 𝑦 𝜓
9 7 8 nfbi ⊢ Ⅎ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
10 opeq12 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
11 copsexgw ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
12 11 eqcoms ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
13 10 12 syl ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
14 13 adantl ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
15 2sp ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) )
16 15 imp ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜑 ↔ 𝜓 ) )
17 14 16 bitr3d ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
18 17 ex ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) ) )
19 5 9 18 exlimd ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) ) )
20 1 4 19 exlimd ⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) ) )
21 elisset ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
22 elisset ⊢ ( 𝐵 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
23 21 22 anim12i ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
24 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
25 23 24 sylibr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
26 20 25 impel ⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) ) ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )