Metamath Proof Explorer


Theorem raleqbi1dv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) (Proof shortened by Steven Nguyen, 5-May-2023)

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