Theorem rexcom4OLD

Description: Obsolete version of rexcom4 as of 26-Aug-2023. Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	rexcom4OLD	$$⊢\exists x\in A\exists y\phi \leftrightarrow \exists y\exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		rexcom	$$⊢\exists x\in A\exists y\in \mathrm{V}\phi \leftrightarrow \exists y\in \mathrm{V}\exists x\in A\phi$$
2		rexv	$$⊢\exists y\in \mathrm{V}\phi \leftrightarrow \exists y\phi$$
3	2	rexbii	$$⊢\exists x\in A\exists y\in \mathrm{V}\phi \leftrightarrow \exists x\in A\exists y\phi$$
4		rexv	$$⊢\exists y\in \mathrm{V}\exists x\in A\phi \leftrightarrow \exists y\exists x\in A\phi$$
5	1 3 4	3bitr3i	$$⊢\exists x\in A\exists y\phi \leftrightarrow \exists y\exists x\in A\phi$$