Metamath Proof Explorer


Theorem copsex2gOLD

Description: Obsolete version of copsex2g as of 1-Sep-2024. (Contributed by NM, 28-May-1995) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis copsex2g.1 ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( 𝜑𝜓 ) )
Assertion copsex2gOLD ( ( 𝐴𝑉𝐵𝑊 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 copsex2g.1 ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( 𝜑𝜓 ) )
2 elisset ( 𝐴𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
3 elisset ( 𝐵𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
4 exdistrv ( ∃ 𝑥𝑦 ( 𝑥 = 𝐴𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
5 nfe1 𝑥𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
6 nfv 𝑥 𝜓
7 5 6 nfbi 𝑥 ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
8 nfe1 𝑦𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
9 8 nfex 𝑦𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
10 nfv 𝑦 𝜓
11 9 10 nfbi 𝑦 ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
12 opeq12 ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
13 copsexgw ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
14 13 eqcoms ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝜑 ↔ ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
15 12 14 syl ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( 𝜑 ↔ ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
16 15 1 bitr3d ( ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
17 11 16 exlimi ( ∃ 𝑦 ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
18 7 17 exlimi ( ∃ 𝑥𝑦 ( 𝑥 = 𝐴𝑦 = 𝐵 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
19 4 18 sylbir ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
20 2 3 19 syl2an ( ( 𝐴𝑉𝐵𝑊 ) → ( ∃ 𝑥𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )