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Theorem moeu2 37735
Description: Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)
Assertion
Ref Expression
moeu2 (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))

Proof of Theorem moeu2
StepHypRef Expression
1 moeu 2569 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2 imor 850 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
31, 2bitri 275 1 (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 844  wex 1773  ∃*wmo 2524  ∃!weu 2554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-mo 2526  df-eu 2555
This theorem is referenced by:  mopickr  37736
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