Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  nndc Unicode version

Theorem nndc 10722
Description: Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
nndc  |-  -.  -. DECID  ph

Proof of Theorem nndc
StepHypRef Expression
1 nnexmid 10721 . 2  |-  -.  -.  ( ph  \/  -.  ph )
2 df-dc 777 . . 3  |-  (DECID  ph  <->  ( ph  \/  -.  ph ) )
32notbii 627 . 2  |-  ( -. DECID  ph  <->  -.  ( ph  \/  -.  ph ) )
41, 3mtbir 629 1  |-  -.  -. DECID  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by:  dcdc  10723
  Copyright terms: Public domain W3C validator