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	$$⊢\exists !x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		ibar	$$⊢x\in A\to \left(\phi \leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
2	1	bicomd	$$⊢x\in A\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \phi \right)$$
3	2	reubiia	$$⊢\exists !x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\in A\phi$$