Theorem moeuex 2612

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 2599	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2	1	rbaibr 547	1 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  ∃*wmo 2567  ∃!weu 2598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-eu 2599

This theorem is used by:  exeupre2  39179