Theorem eean

Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010) (Revised by Mario Carneiro, 6-Oct-2016)

	Ref	Expression
Hypotheses	eean.1	$$⊢Ⅎy\phi$$
	eean.2	$$⊢Ⅎx\psi$$
Assertion	eean	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \left(\exists x\phi \wedge \exists y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eean.1	$$⊢Ⅎy\phi$$
2		eean.2	$$⊢Ⅎx\psi$$
3	1	19.42	$$⊢\exists y\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists y\psi \right)$$
4	3	exbii	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \exists x\left(\phi \wedge \exists y\psi \right)$$
5	2	nfex	$$⊢Ⅎx\exists y\psi$$
6	5	19.41	$$⊢\exists x\left(\phi \wedge \exists y\psi \right)\leftrightarrow \left(\exists x\phi \wedge \exists y\psi \right)$$
7	4 6	bitri	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\leftrightarrow \left(\exists x\phi \wedge \exists y\psi \right)$$