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Metamath Proof Explorer

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	$⊢ \forall x \in A \forall y \in B \neg φ \leftrightarrow \neg \exists x \in A \exists y \in B φ$

Proof

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