Theorem moeuex 2608

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 2595	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	rbaibr 545	1 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1798  ∃*wmo 2563  ∃!weu 2594

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 209  df-an 400  df-eu 2595

This theorem is referenced by:  exeupre2  38932