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 ) -> ( ph <-> ps ) )
Assertion	opelopab2a	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
2		eleq1	|- ( x = A -> ( x e. C <-> A e. C ) )
3		eleq1	|- ( y = B -> ( y e. D <-> B e. D ) )
4	2 3	bi2anan9	|- ( ( x = A /\ y = B ) -> ( ( x e. C /\ y e. D ) <-> ( A e. C /\ B e. D ) ) )
5	4 1	anbi12d	|- ( ( x = A /\ y = B ) -> ( ( ( x e. C /\ y e. D ) /\ ph ) <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
6	5	opelopabga	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ( ( A e. C /\ B e. D ) /\ ps ) ) )
7	6	bianabs	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ps ) )