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Metamath Proof Explorer

Theorem ralnex3

Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020) (Proof shortened by Wolf Lammen, 18-May-2023)

		Ref	Expression
	Assertion	ralnex3	$⊢ \forall x \in A \forall y \in B \forall z \in C \neg φ \leftrightarrow \neg \exists x \in A \exists y \in B \exists z \in C φ$

Proof

Step	Hyp	Ref	Expression
1		ralnex	$⊢ \forall z \in C \neg φ \leftrightarrow \neg \exists z \in C φ$
2	1	2ralbii	$⊢ \forall x \in A \forall y \in B \forall z \in C \neg φ \leftrightarrow \forall x \in A \forall y \in B \neg \exists z \in C φ$
3		ralnex2	$⊢ \forall x \in A \forall y \in B \neg \exists z \in C φ \leftrightarrow \neg \exists x \in A \exists y \in B \exists z \in C φ$
4	2 3	bitri	$⊢ \forall x \in A \forall y \in B \forall z \in C \neg φ \leftrightarrow \neg \exists x \in A \exists y \in B \exists z \in C φ$