Theorem ralnex

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997) (Proof shortened by BJ, 16-Jul-2021)

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

Proof

Step	Hyp	Ref	Expression
1		raln	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$
2		alnex	$$⊢\forall x\neg \left(x\in A\wedge \phi \right)\leftrightarrow \neg \exists x\left(x\in A\wedge \phi \right)$$
3		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
4	2 3	xchbinxr	$$⊢\forall x\neg \left(x\in A\wedge \phi \right)\leftrightarrow \neg \exists x\in A\phi$$
5	1 4	bitri	$$⊢\forall x\in A\neg \phi \leftrightarrow \neg \exists x\in A\phi$$