Theorem reu5

Description: Restricted uniqueness in terms of "at most one". (Contributed by NM, 23-May-1999) (Revised by NM, 16-Jun-2017)

	Ref	Expression
Assertion	reu5	$$⊢\exists !x\in A\phi \leftrightarrow \left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-eu	$$⊢\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \left(\exists x\left(x\in A\wedge \phi \right)\wedge {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
2		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
3		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
5	3 4	anbi12i	$$⊢\left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)\leftrightarrow \left(\exists x\left(x\in A\wedge \phi \right)\wedge {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
6	1 2 5	3bitr4i	$$⊢\exists !x\in A\phi \leftrightarrow \left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)$$