This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2rexbiia

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004)

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

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