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Metamath Proof Explorer

Theorem rexbii2

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004)

		Ref	Expression
	Hypothesis	rexbii2.1	$⊢ (x \in A \land φ) \leftrightarrow (x \in B \land ψ)$
	Assertion	rexbii2	$⊢ \exists x \in A φ \leftrightarrow \exists x \in B ψ$

Step	Hyp	Ref	Expression
1		rexbii2.1	$⊢ (x \in A \land φ) \leftrightarrow (x \in B \land ψ)$
2	1	exbii	$⊢ \exists x (x \in A \land φ) \leftrightarrow \exists x (x \in B \land ψ)$
3		df-rex	$⊢ \exists x \in A φ \leftrightarrow \exists x (x \in A \land φ)$
4		df-rex	$⊢ \exists x \in B ψ \leftrightarrow \exists x (x \in B \land ψ)$
5	2 3 4	3bitr4i	$⊢ \exists x \in A φ \leftrightarrow \exists x \in B ψ$