Metamath Proof Explorer


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 )