Theorem rexab

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014) (Revised by Mario Carneiro, 3-Sep-2015)

	Ref	Expression
Hypothesis	ralab.1	|- ( y = x -> ( ph <-> ps ) )
Assertion	rexab	|- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Proof

Step	Hyp	Ref	Expression
1		ralab.1	|- ( y = x -> ( ph <-> ps ) )
2		df-rex	|- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
3		vex	|- x e. _V
4	3 1	elab	|- ( x e. { y | ph } <-> ps )
5	4	anbi1i	|- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii	|- ( E. x ( x e. { y | ph } /\ ch ) <-> E. x ( ps /\ ch ) )
7	2 6	bitri	|- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )