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Theorem dcn 745
 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 559 . . . 4 (φ → ¬ ¬ φ)
21orim2i 677 . . 3 ((¬ φ φ) → (¬ φ ¬ ¬ φ))
32orcoms 648 . 2 ((φ ¬ φ) → (¬ φ ¬ ¬ φ))
4 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
5 df-dc 742 . 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 628  DECID wdc 741 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 544  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110  df-dc 742 This theorem is referenced by:  pm5.18dc  776  pm4.67dc  780  pm2.54dc  789  imordc  795  pm4.54dc  804  stabtestimpdc  823  annimdc  844  pm4.55dc  845  pm3.12dc  864  pm3.13dc  865  dn1dc  866  xor3dc  1275  dfbi3dc  1285  dcned  2209
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