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Theorem exmoeu 2665
Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
Assertion
Ref Expression
exmoeu (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem exmoeu
StepHypRef Expression
1 df-mo 2637 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
21biimpi 207 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
32com12 32 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
4 exmo 2657 . . . . 5 (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
54ori 879 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
65con1i 146 . . 3 (¬ ∃*𝑥𝜑 → ∃𝑥𝜑)
7 euex 2656 . . 3 (∃!𝑥𝜑 → ∃𝑥𝜑)
86, 7ja 174 . 2 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑)
93, 8impbii 200 1 (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wex 1859  ∃!weu 2632  ∃*wmo 2633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070
This theorem depends on definitions:  df-bi 198  df-or 866  df-ex 1860  df-eu 2636  df-mo 2637
This theorem is referenced by: (None)
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