Metamath Proof Explorer


Theorem rexeqbidva

Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017)

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

Proof

Step Hyp Ref Expression
1 raleqbidva.1
 |-  ( ph -> A = B )
2 raleqbidva.2
 |-  ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3 2 rexbidva
 |-  ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )
4 1 rexeqdv
 |-  ( ph -> ( E. x e. A ch <-> E. x e. B ch ) )
5 3 4 bitrd
 |-  ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )