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Metamath Proof Explorer

Theorem alexn

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

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

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