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

Theorem nbbndc 1328
Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
nbbndc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  <->  ps )  <->  -.  ( ph  <->  ps ) ) ) )

Proof of Theorem nbbndc
StepHypRef Expression
1 xor3dc 1321 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )
21imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  <->  -. 
ps ) ) )
3 con2bidc 805 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) ) ) )
43imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  -. 
ps )  <->  ( ps  <->  -. 
ph ) ) )
52, 4bitrd 186 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ps  <->  -. 
ph ) ) )
6 bicom 138 . . 3  |-  ( ( ps  <->  -.  ph )  <->  ( -.  ph  <->  ps ) )
75, 6syl6rbb 195 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph 
<->  ps )  <->  -.  ( ph 
<->  ps ) ) )
87ex 113 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  <->  ps )  <->  -.  ( ph  <->  ps ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 778
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 779
This theorem is referenced by:  biassdc  1329
  Copyright terms: Public domain W3C validator