Theorem moeuex

Description: Uniqueness implies that existence is equivalent to unique existence. (Contributed by BJ, 7-Oct-2022)

	Ref	Expression
Assertion	moeuex	$$⊢{\exists }^{*}x\phi \to \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
2	1	rbaibr	$$⊢{\exists }^{*}x\phi \to \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$