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

Metamath Proof Explorer

Theorem reuanid

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018) (Proof shortened by Wolf Lammen, 12-Jan-2025)

		Ref	Expression
	Assertion	reuanid	$⊢ \exists! x \in A (x \in A \land φ) \leftrightarrow \exists! x \in A φ$

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	reubiia	$⊢ \exists! x \in A (x \in A \land φ) \leftrightarrow \exists! x \in A φ$