Theorem moeuex 2571

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

Assertion

Ref	Expression
moeuex	⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Proof of Theorem moeuex

Step	Hyp	Ref	Expression
1		df-eu 2558	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	rbaibr 537	1 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1774  ∃*wmo 2527  ∃!weu 2557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-eu 2558

This theorem is referenced by: (None)