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Metamath Proof Explorer

Theorem rmoanid

Description: Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018) (Proof shortened by Wolf Lammen, 12-Jan-2025)

		Ref	Expression
	Assertion	rmoanid	$⊢ \exists^{} x \in A (x \in A \land φ) \leftrightarrow \exists^{} x \in A φ$

Proof

Step	Hyp	Ref	Expression
1		ibar	$⊢ x \in A \to (φ \leftrightarrow (x \in A \land φ))$
2	1	bicomd	$⊢ x \in A \to ((x \in A \land φ) \leftrightarrow φ)$
3	2	rmobiia	$⊢ \exists^{} x \in A (x \in A \land φ) \leftrightarrow \exists^{} x \in A φ$