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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	⊢ ( ∃! 𝑥 𝜑 → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-eu	⊢ ( ∃! 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 ∧ ∃* 𝑥 𝜑 ) )
2	1	simplbi	⊢ ( ∃! 𝑥 𝜑 → ∃ 𝑥 𝜑 )