Theorem rexcom

Description: Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995) (Revised by Mario Carneiro, 14-Oct-2016) (Proof shortened by BJ, 26-Aug-2023) (Proof shortened by Wolf Lammen, 8-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ralcom	$$⊢\forall x\in A\forall y\in B\neg \phi \leftrightarrow \forall y\in B\forall x\in A\neg \phi$$
2		ralnex2	$$⊢\forall x\in A\forall y\in B\neg \phi \leftrightarrow \neg \exists x\in A\exists y\in B\phi$$
3		ralnex2	$$⊢\forall y\in B\forall x\in A\neg \phi \leftrightarrow \neg \exists y\in B\exists x\in A\phi$$
4	1 2 3	3bitr3i	$$⊢\neg \exists x\in A\exists y\in B\phi \leftrightarrow \neg \exists y\in B\exists x\in A\phi$$
5	4	con4bii	$$⊢\exists x\in A\exists y\in B\phi \leftrightarrow \exists y\in B\exists x\in A\phi$$