Metamath Proof Explorer

Theorem exmoeub

Description: Existence implies that uniqueness is equivalent to unique existence. (Contributed by NM, 5-Apr-2004)

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

Proof

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