Theorem alexn

Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Assertion	alexn	$$⊢\forall x\exists y\neg \phi \leftrightarrow \neg \exists x\forall y\phi$$

Proof

Step	Hyp	Ref	Expression
1		exnal	$$⊢\exists y\neg \phi \leftrightarrow \neg \forall y\phi$$
2	1	albii	$$⊢\forall x\exists y\neg \phi \leftrightarrow \forall x\neg \forall y\phi$$
3		alnex	$$⊢\forall x\neg \forall y\phi \leftrightarrow \neg \exists x\forall y\phi$$
4	2 3	bitri	$$⊢\forall x\exists y\neg \phi \leftrightarrow \neg \exists x\forall y\phi$$