Theorem rspcimedv

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	rspcimdv.1	|- ( ph -> A e. B )
	rspcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
Assertion	rspcimedv	|- ( ph -> ( ch -> E. x e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		rspcimdv.1	|- ( ph -> A e. B )
2		rspcimedv.2	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
3	2	con3d	|- ( ( ph /\ x = A ) -> ( -. ps -> -. ch ) )
4	1 3	rspcimdv	|- ( ph -> ( A. x e. B -. ps -> -. ch ) )
5	4	con2d	|- ( ph -> ( ch -> -. A. x e. B -. ps ) )
6		dfrex2	|- ( E. x e. B ps <-> -. A. x e. B -. ps )
7	5 6	syl6ibr	|- ( ph -> ( ch -> E. x e. B ps ) )