Theorem reuanid

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018)

	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		anabs5	$$⊢\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow \left(x\in A\wedge \phi \right)$$
2	1	eubii	$$⊢\exists !x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
3		df-reu	$$⊢\exists !x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)$$
4		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
5	2 3 4	3bitr4i	$$⊢\exists !x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\in A\phi$$