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Theorem exmodc 1932
Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
Assertion
Ref Expression
exmodc (DECID xφ → (xφ ∃*xφ))

Proof of Theorem exmodc
StepHypRef Expression
1 df-dc 734 . 2 (DECID xφ ↔ (xφ ¬ xφ))
2 pm2.21 535 . . . 4 xφ → (xφ∃!xφ))
3 df-mo 1886 . . . 4 (∃*xφ ↔ (xφ∃!xφ))
42, 3sylibr 137 . . 3 xφ∃*xφ)
54orim2i 665 . 2 ((xφ ¬ xφ) → (xφ ∃*xφ))
61, 5sylbi 114 1 (DECID xφ → (xφ ∃*xφ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616  DECID wdc 733  wex 1362  ∃!weu 1882  ∃*wmo 1883
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 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734  df-mo 1886
This theorem is referenced by: (None)
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