Theorem ralnex2

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019) (Proof shortened by Wolf Lammen, 18-May-2023)

	Ref	Expression
Assertion	ralnex2	$$⊢\forall x\in A\forall y\in B\neg \phi \leftrightarrow \neg \exists x\in A\exists y\in B\phi$$

Proof

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