Theorem brabg

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopabg.1	|- ( x = A -> ( ph <-> ps ) )
	opelopabg.2	|- ( y = B -> ( ps <-> ch ) )
	brabg.5	|- R = { <. x , y >. | ph }
Assertion	brabg	|- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		opelopabg.1	|- ( x = A -> ( ph <-> ps ) )
2		opelopabg.2	|- ( y = B -> ( ps <-> ch ) )
3		brabg.5	|- R = { <. x , y >. | ph }
4	1 2	sylan9bb	|- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
5	4 3	brabga	|- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )