Theorem eqbrrdv

Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	eqbrrdv.1	|- ( ph -> Rel A )
	eqbrrdv.2	|- ( ph -> Rel B )
	eqbrrdv.3	|- ( ph -> ( x A y <-> x B y ) )
Assertion	eqbrrdv	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		eqbrrdv.1	|- ( ph -> Rel A )
2		eqbrrdv.2	|- ( ph -> Rel B )
3		eqbrrdv.3	|- ( ph -> ( x A y <-> x B y ) )
4		df-br	|- ( x A y <-> <. x , y >. e. A )
5		df-br	|- ( x B y <-> <. x , y >. e. B )
6	3 4 5	3bitr3g	|- ( ph -> ( <. x , y >. e. A <-> <. x , y >. e. B ) )
7	6	alrimivv	|- ( ph -> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) )
8		eqrel	|- ( ( Rel A /\ Rel B ) -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
9	1 2 8	syl2anc	|- ( ph -> ( A = B <-> A. x A. y ( <. x , y >. e. A <-> <. x , y >. e. B ) ) )
10	7 9	mpbird	|- ( ph -> A = B )