ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xor2dc Unicode version

Theorem xor2dc 1297
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
xor2dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) ) ) )

Proof of Theorem xor2dc
StepHypRef Expression
1 xor3dc 1294 . . . 4  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )
21imp 119 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  <->  -. 
ps ) ) )
3 pm5.17dc 821 . . . 4  |-  (DECID  ps  ->  ( ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) )  <-> 
( ph  <->  -.  ps )
) )
43adantl 266 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  \/  ps )  /\  -.  ( ph  /\  ps ) )  <->  ( ph  <->  -. 
ps ) ) )
52, 4bitr4d 184 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) ) )
65ex 112 1  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  xornbidc  1298
  Copyright terms: Public domain W3C validator