Theorem raleqtrrdv

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

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

Proof

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