Theorem rexeqdv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007)

	Ref	Expression
Hypothesis	raleqdv.1	$$⊢\phi \to A=B$$
Assertion	rexeqdv	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqdv.1	$$⊢\phi \to A=B$$
2		rexeq	$$⊢A=B\to \left(\exists x\in A\psi \leftrightarrow \exists x\in B\psi \right)$$
3	1 2	syl	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in B\psi \right)$$