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	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	raleqbi1dv	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqbi1dv.1	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
2		id	$$⊢A=B\to A=B$$
3	2 1	raleqbidv	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\psi \right)$$