Theorem rexbiia

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999)

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

Proof

Step	Hyp	Ref	Expression
1		rexbiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	rexbii2	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in A\psi$$