Theorem opelopab2a

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypothesis	opelopabga.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	opelopab2a	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
3		eleq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷 ) )
4	2 3	bi2anan9	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ) )
5	4 1	anbi12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) ) )
6	5	opelopabga	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) ) )
7	6	bianabs	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ 𝜓 ) )