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Theorem dcn 737
Description: A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
Assertion
Ref Expression
dcn (DECID φDECID ¬ φ)

Proof of Theorem dcn
StepHypRef Expression
1 notnot1 547 . . . 4 (φ → ¬ ¬ φ)
21orim2i 665 . . 3 ((¬ φ φ) → (¬ φ ¬ ¬ φ))
32orcoms 636 . 2 ((φ ¬ φ) → (¬ φ ¬ ¬ φ))
4 df-dc 734 . 2 (DECID φ ↔ (φ ¬ φ))
5 df-dc 734 . 2 (DECID ¬ φ ↔ (¬ φ ¬ ¬ φ))
63, 4, 53imtr4i 190 1 (DECID φDECID ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 616  DECID wdc 733
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-in1 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734
This theorem is referenced by:  pm5.18dc  770  pm4.67dc  774  pm2.54dc  783  imordc  789  pm4.54dc  798  annimdc  833  pm4.55dc  834  pm3.12dc  853  pm3.13dc  854  dn1dc  855  xor3dc  1261  dfbi3dc  1271
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