Theorem rspcedv

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007) (Revised by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	|- ( ph -> A e. B )
2		rspcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	2	biimprd	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
4	1 3	rspcimedv	|- ( ph -> ( ch -> E. x e. B ps ) )