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Metamath Proof Explorer

Theorem alex

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex . See also the dual pair alnex / exnal . Theorem 19.6 of Margaris p. 89. (Contributed by NM, 12-Mar-1993)

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

Proof

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