Theorem rexeqtrrdv

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

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

Proof

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