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Metamath Proof Explorer

Theorem dfrex2

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by Wolf Lammen, 26-Nov-2019)

		Ref	Expression
	Assertion	dfrex2	$⊢ \exists x \in A φ \leftrightarrow \neg \forall x \in A \neg φ$

Proof

Step	Hyp	Ref	Expression
1		ralnex	$⊢ \forall x \in A \neg φ \leftrightarrow \neg \exists x \in A φ$
2	1	con2bii	$⊢ \exists x \in A φ \leftrightarrow \neg \forall x \in A \neg φ$