Theorem raleqtrdv

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025)

	Ref	Expression
Hypotheses	raleqtrdv.1	$$⊢\phi \to \forall x\in A\psi$$
	raleqtrdv.2	$$⊢\phi \to A=B$$
Assertion	raleqtrdv	$$⊢\phi \to \forall x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		raleqtrdv.1	$$⊢\phi \to \forall x\in A\psi$$
2		raleqtrdv.2	$$⊢\phi \to A=B$$
3	2	raleqdv	$$⊢\phi \to \left(\forall x\in A\psi \leftrightarrow \forall x\in B\psi \right)$$
4	1 3	mpbid	$$⊢\phi \to \forall x\in B\psi$$