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Metamath Proof Explorer

Theorem r2exf

Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex . (Contributed by Mario Carneiro, 14-Oct-2016) Use r2exlem . (Revised by Wolf Lammen, 10-Jan-2020)

		Ref	Expression
	Hypothesis	r2exf.1	$⊢ \underline{Ⅎ} y A$
	Assertion	r2exf	$⊢ \exists x \in A \exists y \in B φ \leftrightarrow \exists x \exists y ((x \in A \land y \in B) \land φ)$

Proof

Step	Hyp	Ref	Expression
1		r2exf.1	$⊢ \underline{Ⅎ} y A$
2	1	r2alf	$⊢ \forall x \in A \forall y \in B \neg φ \leftrightarrow \forall x \forall y ((x \in A \land y \in B) \to \neg φ)$
3	2	r2exlem	$⊢ \exists x \in A \exists y \in B φ \leftrightarrow \exists x \exists y ((x \in A \land y \in B) \land φ)$