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

Theorem dcnn 843
Description: Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 837. The relation between dcn 837 and dcnn 843 is analogous to that between notnot 624 and notnotnot 629 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 843 means that a proposition is testable if and only if its negation is testable, and dcn 837 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
Assertion
Ref Expression
dcnn  |-  (DECID  -.  ph  <-> DECID  -.  -.  ph )

Proof of Theorem dcnn
StepHypRef Expression
1 dcn 837 . 2  |-  (DECID  -.  ph  -> DECID  -. 
-.  ph )
2 stabnot 828 . . 3  |- STAB  -.  ph
3 stdcn 842 . . 3  |-  (STAB  -.  ph  <->  (DECID  -. 
-.  ph  -> DECID  -.  ph ) )
42, 3mpbi 144 . 2  |-  (DECID  -.  -.  ph 
-> DECID  -. 
ph )
51, 4impbii 125 1  |-  (DECID  -.  ph  <-> DECID  -.  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104  STAB wstab 825  DECID wdc 829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator