Theorem rexbii

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994) (Revised by Mario Carneiro, 17-Oct-2016) (Proof shortened by Wolf Lammen, 6-Dec-2019)

	Ref	Expression
Hypothesis	rexbii.1	$$⊢\phi \leftrightarrow \psi$$
Assertion	rexbii	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		rexbii.1	$$⊢\phi \leftrightarrow \psi$$
2	1	anbi2i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	rexbii2	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in A\psi$$