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	$$⊢\forall x\phi \leftrightarrow \neg \exists x\neg \phi$$

Proof

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