Theorem opelopabga

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypothesis	opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
Assertion	opelopabga	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
2		elopab	|- ( <. A , B >. e. { <. x , y >. | ph } <-> E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) )
3	1	copsex2g	|- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) )
4	2 3	bitrid	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )