Metamath Proof Explorer


Theorem cbvrexdva

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypothesis cbvraldva.1
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion cbvrexdva
|- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )

Proof

Step Hyp Ref Expression
1 cbvraldva.1
 |-  ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2 eqidd
 |-  ( ( ph /\ x = y ) -> A = A )
3 1 2 cbvrexdva2
 |-  ( ph -> ( E. x e. A ps <-> E. y e. A ch ) )