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Metamath Proof Explorer

Theorem abn0

Description: Nonempty class abstraction. See also ab0 . (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016) Avoid df-clel , ax-8 . (Revised by GG, 30-Aug-2024)

		Ref	Expression
	Assertion	abn0	$⊢ \{x \| φ\} \neq \emptyset \leftrightarrow \exists x φ$

Proof

Step	Hyp	Ref	Expression
1		ab0	$⊢ \{x \| φ\} = \emptyset \leftrightarrow \forall x \neg φ$
2	1	notbii	$⊢ \neg \{x \| φ\} = \emptyset \leftrightarrow \neg \forall x \neg φ$
3		df-ne	$⊢ \{x \| φ\} \neq \emptyset \leftrightarrow \neg \{x \| φ\} = \emptyset$
4		df-ex	$⊢ \exists x φ \leftrightarrow \neg \forall x \neg φ$
5	2 3 4	3bitr4i	$⊢ \{x \| φ\} \neq \emptyset \leftrightarrow \exists x φ$