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Metamath Proof Explorer

Theorem exnal

Description: Existential quantification of negation is equivalent to negation of universal quantification. Dual of alnex . See also the dual pair df-ex / alex . Theorem 19.14 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

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

Proof

Step	Hyp	Ref	Expression
1		alex	$⊢ \forall x φ \leftrightarrow \neg \exists x \neg φ$
2	1	con2bii	$⊢ \exists x \neg φ \leftrightarrow \neg \forall x φ$