Theorem rexcom4

Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019)

	Ref	Expression
Assertion	rexcom4	$$⊢\exists x\in A\exists y\phi \leftrightarrow \exists y\exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-rex	$$⊢\exists x\in A\exists y\phi \leftrightarrow \exists x\left(x\in A\wedge \exists y\phi \right)$$
2		19.42v	$$⊢\exists y\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \exists y\phi \right)$$
3	2	bicomi	$$⊢\left(x\in A\wedge \exists y\phi \right)\leftrightarrow \exists y\left(x\in A\wedge \phi \right)$$
4	3	exbii	$$⊢\exists x\left(x\in A\wedge \exists y\phi \right)\leftrightarrow \exists x\exists y\left(x\in A\wedge \phi \right)$$
5		excom	$$⊢\exists x\exists y\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\exists x\left(x\in A\wedge \phi \right)$$
6		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
7	6	bicomi	$$⊢\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\in A\phi$$
8	7	exbii	$$⊢\exists y\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\exists x\in A\phi$$
9	5 8	bitri	$$⊢\exists x\exists y\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\exists x\in A\phi$$
10	4 9	bitri	$$⊢\exists x\left(x\in A\wedge \exists y\phi \right)\leftrightarrow \exists y\exists x\in A\phi$$
11	1 10	bitri	$$⊢\exists x\in A\exists y\phi \leftrightarrow \exists y\exists x\in A\phi$$