Theorem rexbii2

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004)

	Ref	Expression
Hypothesis	rexbii2.1	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \psi \right)$$
Assertion	rexbii2	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		rexbii2.1	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \psi \right)$$
2	1	exbii	$$⊢\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\left(x\in B\wedge \psi \right)$$
3		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
4		df-rex	$$⊢\exists x\in B\psi \leftrightarrow \exists x\left(x\in B\wedge \psi \right)$$
5	2 3 4	3bitr4i	$$⊢\exists x\in A\phi \leftrightarrow \exists x\in B\psi$$