Theorem copsex2gd

Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995) Use a similar proof to copsex4g to reduce axiom usage. (Revised by SN, 1-Sep-2024) Adapt copsex2g $p to deduction form. (Revised by BJ, 28-Mar-2026) Do not use copsex2g . (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	copsex2gd.is	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	copsex2gd	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2gd.is	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
2		eqcom	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
3		vex	⊢ 𝑥 ∈ V
4		vex	⊢ 𝑦 ∈ V
5	3 4	opth	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
6	2 5	bitri	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
7	6	anbi1i	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜓 ) )
8	7	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜓 ) )
9		simpr	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
10	9 1	cgsex2gd	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ 𝜓 ) ↔ 𝜒 ) )
11	8 10	bitrid	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )