Theorem euex

Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993) (Proof shortened by Andrew Salmon, 9-Jul-2011) (Proof shortened by Wolf Lammen, 4-Dec-2018) (Proof shortened by BJ, 7-Oct-2022)

	Ref	Expression
Assertion	euex	$$⊢\exists !x\phi \to \exists x\phi$$

Proof

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