Metamath Proof Explorer


Theorem raleq

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023)

Ref Expression
Assertion raleq
|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )

Proof

Step Hyp Ref Expression
1 biidd
 |-  ( A = B -> ( ph <-> ph ) )
2 1 raleqbi1dv
 |-  ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )