Metamath Proof Explorer


Theorem rexeqbi1dv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997) (Proof shortened by Steven Nguyen, 5-May-2023)

Ref Expression
Hypothesis raleqbi1dv.1
|- ( A = B -> ( ph <-> ps ) )
Assertion rexeqbi1dv
|- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )

Proof

Step Hyp Ref Expression
1 raleqbi1dv.1
 |-  ( A = B -> ( ph <-> ps ) )
2 id
 |-  ( A = B -> A = B )
3 2 1 rexeqbidvv
 |-  ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) )