Theorem rexnal2

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	rexnal2	$$⊢\exists x\in A\exists y\in B\neg \phi \leftrightarrow \neg \forall x\in A\forall y\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		rexnal	$$⊢\exists y\in B\neg \phi \leftrightarrow \neg \forall y\in B\phi$$
2	1	rexbii	$$⊢\exists x\in A\exists y\in B\neg \phi \leftrightarrow \exists x\in A\neg \forall y\in B\phi$$
3		rexnal	$$⊢\exists x\in A\neg \forall y\in B\phi \leftrightarrow \neg \forall x\in A\forall y\in B\phi$$
4	2 3	bitri	$$⊢\exists x\in A\exists y\in B\neg \phi \leftrightarrow \neg \forall x\in A\forall y\in B\phi$$