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Metamath Proof Explorer

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	$⊢ \exists! x φ \to \exists x φ$

Proof

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