Theorem alnex

Description: Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal (but does not depend on ax-4 contrary to it). See also the dual pair df-ex / alex . Theorem 19.7 of Margaris p. 89. (Contributed by NM, 12-Mar-1993)

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

Proof

Step	Hyp	Ref	Expression
1		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
2	1	con2bii	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$