Theorem rexanid

Description: Cancellation law for restricted existential quantification. (Contributed by Peter Mazsa, 24-May-2018) (Proof shortened by Wolf Lammen, 8-Jul-2023)

	Ref	Expression
Assertion	rexanid	$$⊢\exists x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\in A\phi$$

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