Theorem opelopabg

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 28-May-1995) (Revised by Mario Carneiro, 19-Dec-2013)

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

Proof

Step	Hyp	Ref	Expression
1		opelopabg.1	|- ( x = A -> ( ph <-> ps ) )
2		opelopabg.2	|- ( y = B -> ( ps <-> ch ) )
3	1 2	sylan9bb	|- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4	3	opelopabga	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )