intexrab

Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003)

		Ref	Expression
	Assertion	intexrab	$⊢ \exists x \in A φ \leftrightarrow ⋂ \{x \in A \| φ\} \in V$

Step	Hyp	Ref	Expression
1		intexab	$⊢ \exists x (x \in A \land φ) \leftrightarrow ⋂ \{x \| (x \in A \land φ)\} \in V$
2		df-rex	$⊢ \exists x \in A φ \leftrightarrow \exists x (x \in A \land φ)$
3		df-rab	$⊢ \{x \in A \| φ\} = \{x \| (x \in A \land φ)\}$
4	3	inteqi	$⊢ ⋂ \{x \in A \| φ\} = ⋂ \{x \| (x \in A \land φ)\}$
5	4	eleq1i	$⊢ ⋂ \{x \in A \| φ\} \in V \leftrightarrow ⋂ \{x \| (x \in A \land φ)\} \in V$
6	1 2 5	3bitr4i	$⊢ \exists x \in A φ \leftrightarrow ⋂ \{x \in A \| φ\} \in V$

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