Theorem 2rexbiia

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004)

	Ref	Expression
Hypothesis	2rexbiia.1	$$⊢\left(x\in A\wedge y\in B\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	2rexbiia	$$⊢\exists x\in A\exists y\in B\phi \leftrightarrow \exists x\in A\exists y\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		2rexbiia.1	$$⊢\left(x\in A\wedge y\in B\right)\to \left(\phi \leftrightarrow \psi \right)$$
2	1	rexbidva	$$⊢x\in A\to \left(\exists y\in B\phi \leftrightarrow \exists y\in B\psi \right)$$
3	2	rexbiia	$$⊢\exists x\in A\exists y\in B\phi \leftrightarrow \exists x\in A\exists y\in B\psi$$