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Metamath Proof Explorer

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	$⊢ \forall x \in A φ \leftrightarrow \neg \exists x \in A \neg φ$

Proof

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