Theorem dfral2

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) Allow shortening of rexnal . (Revised by Wolf Lammen, 9-Dec-2019)

	Ref	Expression
Assertion	dfral2	$$⊢\forall x\in A\phi \leftrightarrow \neg \exists x\in A\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		notnotb	$$⊢\phi \leftrightarrow \neg \neg \phi$$
2	1	ralbii	$$⊢\forall x\in A\phi \leftrightarrow \forall x\in A\neg \neg \phi$$
3		ralnex	$$⊢\forall x\in A\neg \neg \phi \leftrightarrow \neg \exists x\in A\neg \phi$$
4	2 3	bitri	$$⊢\forall x\in A\phi \leftrightarrow \neg \exists x\in A\neg \phi$$