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Theorem exmoeudc 2038
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 1979 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
21biimpi 119 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
32com12 30 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
41biimpri 132 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑)
5 euex 2005 . . . 4 (∃!𝑥𝜑 → ∃𝑥𝜑)
64, 5imim12i 59 . . 3 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7 peircedc 882 . . 3 (DECID𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑))
86, 7syl5 32 . 2 (DECID𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
93, 8impbid2 142 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  DECID wdc 802  wex 1451  ∃!weu 1975  ∃*wmo 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-dc 803  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by: (None)
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