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Theorem exmoeudc 2011
Description: Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
Assertion
Ref Expression
exmoeudc (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))

Proof of Theorem exmoeudc
StepHypRef Expression
1 df-mo 1952 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
21biimpi 118 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
32com12 30 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
41biimpri 131 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑)
5 euex 1978 . . . 4 (∃!𝑥𝜑 → ∃𝑥𝜑)
64, 5imim12i 58 . . 3 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7 peircedc 858 . . 3 (DECID𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑))
86, 7syl5 32 . 2 (DECID𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
93, 8impbid2 141 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  DECID wdc 780  wex 1426  ∃!weu 1948  ∃*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-dc 781  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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