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Theorem exmoeudc 1979
 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 1920 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
21biimpi 117 . . 3 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
32com12 30 . 2 (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑))
41biimpri 128 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃*𝑥𝜑)
5 euex 1946 . . . 4 (∃!𝑥𝜑 → ∃𝑥𝜑)
64, 5imim12i 57 . . 3 ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
7 peircedc 831 . . 3 (DECID𝑥𝜑 → (((∃𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) → ∃𝑥𝜑))
86, 7syl5 32 . 2 (DECID𝑥𝜑 → ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑))
93, 8impbid2 135 1 (DECID𝑥𝜑 → (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  DECID wdc 753  ∃wex 1397  ∃!weu 1916  ∃*wmo 1917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-dc 754  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920 This theorem is referenced by: (None)
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