Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dcn GIF version

Theorem dcn 746
 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 notnot 559 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21orim2i 678 . . 3 ((¬ 𝜑𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32orcoms 649 . 2 ((𝜑 ∨ ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4 df-dc 743 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 743 . 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 629  DECID wdc 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  pm5.18dc  777  pm4.67dc  781  pm2.54dc  790  imordc  796  pm4.54dc  805  stabtestimpdc  824  annimdc  845  pm4.55dc  846  pm3.12dc  865  pm3.13dc  866  dn1dc  867  xor3dc  1278  dfbi3dc  1288  dcned  2212
 Copyright terms: Public domain W3C validator