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

Theorem xordidc 1306
Description: Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)
Assertion
Ref Expression
xordidc  |-  (DECID  ph  ->  (DECID  ps 
->  (DECID  ch  ->  ( ( ph  /\  ( ps  \/_  ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) ) ) ) )

Proof of Theorem xordidc
StepHypRef Expression
1 dcbi 855 . . . . 5  |-  (DECID  ps  ->  (DECID  ch 
-> DECID  ( ps  <->  ch ) ) )
21imp 119 . . . 4  |-  ( (DECID  ps 
/\ DECID  ch )  -> DECID  ( ps  <->  ch )
)
3 annimdc 856 . . . . . 6  |-  (DECID  ph  ->  (DECID  ( ps  <->  ch )  ->  (
( ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  ( ph  ->  ( ps  <->  ch )
) ) ) )
43imp 119 . . . . 5  |-  ( (DECID  ph  /\ DECID  ( ps  <->  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ph  ->  ( ps  <->  ch )
) ) )
5 pm5.32 434 . . . . . 6  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
65notbii 604 . . . . 5  |-  ( -.  ( ph  ->  ( ps 
<->  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
74, 6syl6bb 189 . . . 4  |-  ( (DECID  ph  /\ DECID  ( ps  <->  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) ) )
82, 7sylan2 274 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) ) )
9 xornbidc 1298 . . . . . 6  |-  (DECID  ps  ->  (DECID  ch 
->  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch ) ) ) )
109imp 119 . . . . 5  |-  ( (DECID  ps 
/\ DECID  ch )  ->  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch )
) )
1110adantl 266 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch ) ) )
1211anbi2d 445 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ph  /\  -.  ( ps  <->  ch )
) ) )
13 dcan 853 . . . . . 6  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  ps )
) )
1413imp 119 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  /\  ps )
)
1514adantrr 456 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> DECID  ( ph  /\  ps ) )
16 dcan 853 . . . . . 6  |-  (DECID  ph  ->  (DECID  ch 
-> DECID  ( ph  /\  ch )
) )
1716imp 119 . . . . 5  |-  ( (DECID  ph  /\ DECID  ch )  -> DECID 
( ph  /\  ch )
)
1817adantrl 455 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> DECID  ( ph  /\  ch ) )
19 xornbidc 1298 . . . 4  |-  (DECID  ( ph  /\ 
ps )  ->  (DECID  ( ph  /\  ch )  -> 
( ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) )  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\  ch ) ) ) ) )
2015, 18, 19sylc 60 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) )  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\  ch ) ) ) )
218, 12, 203bitr4d 213 . 2  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
2221exp32 351 1  |-  (DECID  ph  ->  (DECID  ps 
->  (DECID  ch  ->  ( ( ph  /\  ( ps  \/_  ch ) )  <->  ( ( ph  /\  ps )  \/_  ( ph  /\  ch )
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102  DECID wdc 753    \/_ wxo 1282
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  df-xor 1283
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator