Theorem opelopabb

Description: Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023)

	Ref	Expression
Hypotheses	opelopabb.xph	|- ( ph -> A. x ph )
	opelopabb.yph	|- ( ph -> A. y ph )
	opelopabb.xch	|- ( ph -> F/ x ch )
	opelopabb.ych	|- ( ph -> F/ y ch )
	opelopabb.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
Assertion	opelopabb	|- ( ph -> ( <. A , B >. e. { <. x , y >. | ps } <-> ( ( A e. _V /\ B e. _V ) /\ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabb.xph	|- ( ph -> A. x ph )
2		opelopabb.yph	|- ( ph -> A. y ph )
3		opelopabb.xch	|- ( ph -> F/ x ch )
4		opelopabb.ych	|- ( ph -> F/ y ch )
5		opelopabb.is	|- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )
6		elopab	|- ( <. A , B >. e. { <. x , y >. | ps } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) )
7	1 2 3 4 5	copsex2b	|- ( ph -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps ) <-> ( ( A e. _V /\ B e. _V ) /\ ch ) ) )
8	6 7	syl5bb	|- ( ph -> ( <. A , B >. e. { <. x , y >. | ps } <-> ( ( A e. _V /\ B e. _V ) /\ ch ) ) )