Theorem abn0

Description: Nonempty class abstraction. See also ab0 . (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016)

	Ref	Expression
Assertion	abn0	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfab1	$$⊢\underset{\_}{Ⅎ}x\left\{x|\phi \right\}$$
2	1	n0f	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \exists xx\in \left\{x|\phi \right\}$$
3		abid	$$⊢x\in \left\{x|\phi \right\}\leftrightarrow \phi$$
4	3	exbii	$$⊢\exists xx\in \left\{x|\phi \right\}\leftrightarrow \exists x\phi$$
5	2 4	bitri	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \exists x\phi$$