Theorem opelopab2

Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopab2.1	|- ( x = A -> ( ph <-> ps ) )
	opelopab2.2	|- ( y = B -> ( ps <-> ch ) )
Assertion	opelopab2	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		opelopab2.1	|- ( x = A -> ( ph <-> ps ) )
2		opelopab2.2	|- ( y = B -> ( ps <-> ch ) )
3	1 2	sylan9bb	|- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4	3	opelopab2a	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } <-> ch ) )