Theorem eqrelrdv2

Description: A version of eqrelrdv . (Contributed by Rodolfo Medina, 10-Oct-2010)

	Ref	Expression
Hypothesis	eqrelrdv2.1	|- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
Assertion	eqrelrdv2	|- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B )

Proof

Step	Hyp	Ref	Expression
1		eqrelrdv2.1	|- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
2	1	alrimivv	|- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
3		eqrel	|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
4	2 3	syl5ibr	|- ( ( Rel A /\ Rel B ) -> ( ph -> A = B ) )
5	4	imp	|- ( ( ( Rel A /\ Rel B ) /\ ph ) -> A = B )