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Theorem exmoeudc 1960
 Description: Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
Assertion
Ref Expression
exmoeudc (DECID xφ → (xφ ↔ (∃*xφ∃!xφ)))

Proof of Theorem exmoeudc
StepHypRef Expression
1 df-mo 1901 . . . 4 (∃*xφ ↔ (xφ∃!xφ))
21biimpi 113 . . 3 (∃*xφ → (xφ∃!xφ))
32com12 27 . 2 (xφ → (∃*xφ∃!xφ))
41biimpri 124 . . . 4 ((xφ∃!xφ) → ∃*xφ)
5 euex 1927 . . . 4 (∃!xφxφ)
64, 5imim12i 53 . . 3 ((∃*xφ∃!xφ) → ((xφ∃!xφ) → xφ))
7 peircedc 819 . . 3 (DECID xφ → (((xφ∃!xφ) → xφ) → xφ))
86, 7syl5 28 . 2 (DECID xφ → ((∃*xφ∃!xφ) → xφ))
93, 8impbid2 131 1 (DECID xφ → (xφ ↔ (∃*xφ∃!xφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  DECID wdc 741  ∃wex 1378  ∃!weu 1897  ∃*wmo 1898 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-dc 742  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901 This theorem is referenced by: (None)
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