ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.15dc Unicode version

Theorem pm5.15dc 1296
Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
pm5.15dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( ph  <->  ps )  \/  ( ph  <->  -.  ps )
) ) )

Proof of Theorem pm5.15dc
StepHypRef Expression
1 xor3dc 1294 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( ph  <->  ps )  <->  ( ph  <->  -.  ps )
) ) )
21imp 119 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  <->  ( ph  <->  -. 
ps ) ) )
32biimpd 136 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( ph 
<->  ps )  ->  ( ph 
<->  -.  ps ) ) )
4 dcbi 855 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  <->  ps ) ) )
54imp 119 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  <->  ps ) )
6 dfordc 802 . . . 4  |-  (DECID  ( ph  <->  ps )  ->  ( (
( ph  <->  ps )  \/  ( ph 
<->  -.  ps ) )  <-> 
( -.  ( ph  <->  ps )  ->  ( ph  <->  -. 
ps ) ) ) )
75, 6syl 14 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph 
<->  ps )  \/  ( ph 
<->  -.  ps ) )  <-> 
( -.  ( ph  <->  ps )  ->  ( ph  <->  -. 
ps ) ) ) )
83, 7mpbird 160 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  <->  ps )  \/  ( ph  <->  -. 
ps ) ) )
98ex 112 1  |-  (DECID  ph  ->  (DECID  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: (None)
  Copyright terms: Public domain W3C validator