ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm4.55dc Unicode version

Theorem pm4.55dc 880
Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
Assertion
Ref Expression
pm4.55dc  |-  (DECID  ph  ->  (DECID  ps 
->  ( -.  ( -. 
ph  /\  ps )  <->  (
ph  \/  -.  ps )
) ) )

Proof of Theorem pm4.55dc
StepHypRef Expression
1 pm4.54dc 839 . . . . 5  |-  (DECID  ph  ->  (DECID  ps 
->  ( ( -.  ph  /\ 
ps )  <->  -.  ( ph  \/  -.  ps )
) ) )
21imp 122 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( -. 
ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) )
3 dcn 780 . . . . . . . . 9  |-  (DECID  ps  -> DECID  -.  ps )
43anim2i 334 . . . . . . . 8  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID 
ph  /\ DECID  -.  ps ) )
5 dcor 877 . . . . . . . . 9  |-  (DECID  ph  ->  (DECID  -. 
ps  -> DECID 
( ph  \/  -.  ps ) ) )
65imp 122 . . . . . . . 8  |-  ( (DECID  ph  /\ DECID  -.  ps )  -> DECID  ( ph  \/  -.  ps ) )
74, 6syl 14 . . . . . . 7  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  \/  -.  ps ) )
8 dcn 780 . . . . . . . . 9  |-  (DECID  ph  -> DECID  -.  ph )
9 dcan 876 . . . . . . . . 9  |-  (DECID  -.  ph  ->  (DECID  ps  -> DECID  ( -.  ph  /\  ps ) ) )
108, 9syl 14 . . . . . . . 8  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( -.  ph  /\  ps ) ) )
1110imp 122 . . . . . . 7  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( -.  ph  /\  ps ) )
127, 11jca 300 . . . . . 6  |-  ( (DECID  ph  /\ DECID  ps )  ->  (DECID  ( ph  \/  -.  ps )  /\ DECID  ( -.  ph  /\  ps ) ) )
13 con2bidc 803 . . . . . . 7  |-  (DECID  ( ph  \/  -.  ps )  -> 
(DECID  ( -.  ph  /\  ps )  ->  ( ( ( ph  \/  -.  ps )  <->  -.  ( -.  ph 
/\  ps ) )  <->  ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) ) )
1413imp 122 . . . . . 6  |-  ( (DECID  (
ph  \/  -.  ps )  /\ DECID  ( -.  ph  /\  ps ) )  ->  (
( ( ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps )
)  <->  ( ( -. 
ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) )
1512, 14syl 14 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( (
ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps ) )  <->  ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) ) ) )
1615biimprd 156 . . . 4  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -.  ps ) )  ->  (
( ph  \/  -.  ps )  <->  -.  ( -.  ph 
/\  ps ) ) ) )
172, 16mpd 13 . . 3  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( ( ph  \/  -.  ps )  <->  -.  ( -.  ph  /\  ps )
) )
1817bicomd 139 . 2  |-  ( (DECID  ph  /\ DECID  ps )  ->  ( -.  ( -.  ph  /\  ps )  <->  (
ph  \/  -.  ps )
) )
1918ex 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    \/ 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: (None)
  Copyright terms: Public domain W3C validator