Theorem rexeqtrdv

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

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

Proof

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