Theorem rexnal3

Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020)

	Ref	Expression
Assertion	rexnal3	$$⊢\exists x\in A\exists y\in B\exists z\in C\neg \phi \leftrightarrow \neg \forall x\in A\forall y\in B\forall z\in C\phi$$

Proof

Step	Hyp	Ref	Expression
1		rexnal	$$⊢\exists z\in C\neg \phi \leftrightarrow \neg \forall z\in C\phi$$
2	1	2rexbii	$$⊢\exists x\in A\exists y\in B\exists z\in C\neg \phi \leftrightarrow \exists x\in A\exists y\in B\neg \forall z\in C\phi$$
3		rexnal2	$$⊢\exists x\in A\exists y\in B\neg \forall z\in C\phi \leftrightarrow \neg \forall x\in A\forall y\in B\forall z\in C\phi$$
4	2 3	bitri	$$⊢\exists x\in A\exists y\in B\exists z\in C\neg \phi \leftrightarrow \neg \forall x\in A\forall y\in B\forall z\in C\phi$$