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Metamath Proof Explorer

Theorem reubiia

Description: Formula-building rule for restricted existential uniqueness quantifier (inference form). (Contributed by NM, 14-Nov-2004)

		Ref	Expression
	Hypothesis	rmobiia.1	$⊢ x \in A \to (φ \leftrightarrow ψ)$
	Assertion	reubiia	$⊢ \exists! x \in A φ \leftrightarrow \exists! x \in A ψ$

Step	Hyp	Ref	Expression
1		rmobiia.1	$⊢ x \in A \to (φ \leftrightarrow ψ)$
2	1	pm5.32i	$⊢ (x \in A \land φ) \leftrightarrow (x \in A \land ψ)$
3	2	eubii	$⊢ \exists! x (x \in A \land φ) \leftrightarrow \exists! x (x \in A \land ψ)$
4		df-reu	$⊢ \exists! x \in A φ \leftrightarrow \exists! x (x \in A \land φ)$
5		df-reu	$⊢ \exists! x \in A ψ \leftrightarrow \exists! x (x \in A \land ψ)$
6	3 4 5	3bitr4i	$⊢ \exists! x \in A φ \leftrightarrow \exists! x \in A ψ$