Theorem rexnal

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by Wolf Lammen, 9-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		dfral2	$$⊢\forall x\in A\phi \leftrightarrow \neg \exists x\in A\neg \phi$$
2	1	con2bii	$$⊢\exists x\in A\neg \phi \leftrightarrow \neg \forall x\in A\phi$$