Theorem copsex2dv

Description: Implicit substitution deduction for ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025)

	Ref	Expression
Hypotheses	copsex2dv.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	copsex2dv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
	copsex2dv.1	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	copsex2dv	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2dv.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
2		copsex2dv.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
3		copsex2dv.1	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
4	3	ex	⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) )
5	4	alrimivv	⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) )
6		copsex2t	⊢ ( ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) ∧ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )
7	5 1 2 6	syl12anc	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ 𝜒 ) )