Theorem ralab

Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010)

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

Proof

Step	Hyp	Ref	Expression
1		ralab.1	|- ( y = x -> ( ph <-> ps ) )
2		df-ral	|- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
3		vex	|- x e. _V
4	3 1	elab	|- ( x e. { y | ph } <-> ps )
5	4	imbi1i	|- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii	|- ( A. x ( x e. { y | ph } -> ch ) <-> A. x ( ps -> ch ) )
7	2 6	bitri	|- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )