Metamath Proof Explorer

Theorem 2rexbidva

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004)

		Ref	Expression
	Hypothesis	2ralbidva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \leftrightarrow χ)$
	Assertion	2rexbidva	$⊢ φ \to (\exists x \in A \exists y \in B ψ \leftrightarrow \exists x \in A \exists y \in B χ)$

Step	Hyp	Ref	Expression
1		2ralbidva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \leftrightarrow χ)$
2	1	anassrs	$⊢ ((φ \land x \in A) \land y \in B) \to (ψ \leftrightarrow χ)$
3	2	rexbidva	$⊢ (φ \land x \in A) \to (\exists y \in B ψ \leftrightarrow \exists y \in B χ)$
4	3	rexbidva	$⊢ φ \to (\exists x \in A \exists y \in B ψ \leftrightarrow \exists x \in A \exists y \in B χ)$