Theorem eeor

Description: Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994)

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

Proof

Step	Hyp	Ref	Expression
1		eeor.1	$$⊢Ⅎy\phi$$
2		eeor.2	$$⊢Ⅎx\psi$$
3	1	19.45	$$⊢\exists y\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists y\psi \right)$$
4	3	exbii	$$⊢\exists x\exists y\left(\phi \vee \psi \right)\leftrightarrow \exists x\left(\phi \vee \exists y\psi \right)$$
5	2	nfex	$$⊢Ⅎx\exists y\psi$$
6	5	19.44	$$⊢\exists x\left(\phi \vee \exists y\psi \right)\leftrightarrow \left(\exists x\phi \vee \exists y\psi \right)$$
7	4 6	bitri	$$⊢\exists x\exists y\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists y\psi \right)$$