Theorem excom13

Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995)

	Ref	Expression
Assertion	excom13	$$⊢\exists x\exists y\exists z\phi \leftrightarrow \exists z\exists y\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		excom	$$⊢\exists x\exists y\exists z\phi \leftrightarrow \exists y\exists x\exists z\phi$$
2		excom	$$⊢\exists x\exists z\phi \leftrightarrow \exists z\exists x\phi$$
3	2	exbii	$$⊢\exists y\exists x\exists z\phi \leftrightarrow \exists y\exists z\exists x\phi$$
4		excom	$$⊢\exists y\exists z\exists x\phi \leftrightarrow \exists z\exists y\exists x\phi$$
5	1 3 4	3bitri	$$⊢\exists x\exists y\exists z\phi \leftrightarrow \exists z\exists y\exists x\phi$$