Theorem rspcev

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998) Drop ax-10 , ax-11 , ax-12 . (Revised by SN, 12-Dec-2023)

	Ref	Expression
Hypothesis	rspcv.1	|- ( x = A -> ( ph <-> ps ) )
Assertion	rspcev	|- ( ( A e. B /\ ps ) -> E. x e. B ph )

Proof

Step	Hyp	Ref	Expression
1		rspcv.1	|- ( x = A -> ( ph <-> ps ) )
2		id	|- ( A e. B -> A e. B )
3	1	adantl	|- ( ( A e. B /\ x = A ) -> ( ph <-> ps ) )
4	2 3	rspcedv	|- ( A e. B -> ( ps -> E. x e. B ph ) )
5	4	imp	|- ( ( A e. B /\ ps ) -> E. x e. B ph )