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	$$⊢\forall x\in A\forall y\in B\forall z\in C\neg \phi \leftrightarrow \neg \exists x\in A\exists y\in B\exists z\in C\phi$$

Proof

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