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Theorem exmoeu 2246
 Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exmoeu (xφ ↔ (∃*xφ∃!xφ))

Proof of Theorem exmoeu
StepHypRef Expression
1 df-mo 2209 . . . 4 (∃*xφ ↔ (xφ∃!xφ))
21biimpi 186 . . 3 (∃*xφ → (xφ∃!xφ))
32com12 27 . 2 (xφ → (∃*xφ∃!xφ))
41biimpri 197 . . . 4 ((xφ∃!xφ) → ∃*xφ)
5 euex 2227 . . . 4 (∃!xφxφ)
64, 5imim12i 53 . . 3 ((∃*xφ∃!xφ) → ((xφ∃!xφ) → xφ))
7 peirce 172 . . 3 (((xφ∃!xφ) → xφ) → xφ)
86, 7syl 15 . 2 ((∃*xφ∃!xφ) → xφ)
93, 8impbii 180 1 (xφ ↔ (∃*xφ∃!xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
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