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Metamath Proof Explorer

Theorem rabn0

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

		Ref	Expression
	Assertion	rabn0	$⊢ \{x \in A \| φ\} \neq \emptyset \leftrightarrow \exists x \in A φ$

Step	Hyp	Ref	Expression
1		rabeq0	$⊢ \{x \in A \| φ\} = \emptyset \leftrightarrow \forall x \in A \neg φ$
2	1	necon3abii	$⊢ \{x \in A \| φ\} \neq \emptyset \leftrightarrow \neg \forall x \in A \neg φ$
3		dfrex2	$⊢ \exists x \in A φ \leftrightarrow \neg \forall x \in A \neg φ$
4	2 3	bitr4i	$⊢ \{x \in A \| φ\} \neq \emptyset \leftrightarrow \exists x \in A φ$