Theorem rabn0

Description: Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999) (Revised by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	rabn0	$$⊢\left\{x\in A|\phi \right\}\ne \varnothing \leftrightarrow \exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		rabeq0	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \forall x\in A\neg \phi$$
2	1	necon3abii	$$⊢\left\{x\in A|\phi \right\}\ne \varnothing \leftrightarrow \neg \forall x\in A\neg \phi$$
3		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
4	2 3	bitr4i	$$⊢\left\{x\in A|\phi \right\}\ne \varnothing \leftrightarrow \exists x\in A\phi$$