Theorem moeuex

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

	Ref	Expression
Assertion	moeuex	⊢ ( ∃* 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ ∃! 𝑥 𝜑 ) )

Proof

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