Metamath Proof Explorer


Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995)

Ref Expression
Hypotheses ceqsexv.1
|- A e. _V
ceqsexv.2
|- ( x = A -> ( ph <-> ps ) )
Assertion ceqsexv
|- ( E. x ( x = A /\ ph ) <-> ps )

Proof

Step Hyp Ref Expression
1 ceqsexv.1
 |-  A e. _V
2 ceqsexv.2
 |-  ( x = A -> ( ph <-> ps ) )
3 nfv
 |-  F/ x ps
4 3 1 2 ceqsex
 |-  ( E. x ( x = A /\ ph ) <-> ps )