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	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢A=B\to \left(\phi \leftrightarrow \phi \right)$$
2	1	raleqbi1dv	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\phi \right)$$