Metamath Proof Explorer

Theorem rexin

Description: Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018)

		Ref	Expression
	Assertion	rexin	$⊢ \exists x \in (A \cap B) φ \leftrightarrow \exists x \in A (x \in B \land φ)$

Step	Hyp	Ref	Expression
1		elin	$⊢ x \in (A \cap B) \leftrightarrow (x \in A \land x \in B)$
2	1	anbi1i	$⊢ (x \in (A \cap B) \land φ) \leftrightarrow ((x \in A \land x \in B) \land φ)$
3		anass	$⊢ ((x \in A \land x \in B) \land φ) \leftrightarrow (x \in A \land (x \in B \land φ))$
4	2 3	bitri	$⊢ (x \in (A \cap B) \land φ) \leftrightarrow (x \in A \land (x \in B \land φ))$
5	4	rexbii2	$⊢ \exists x \in (A \cap B) φ \leftrightarrow \exists x \in A (x \in B \land φ)$