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Theorem xordidc 1335
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 882 . . . . 5  |-  (DECID  ps  ->  (DECID  ch 
-> DECID  ( ps  <->  ch ) ) )
21imp 122 . . . 4  |-  ( (DECID  ps 
/\ DECID  ch )  -> DECID  ( ps  <->  ch )
)
3 annimdc 883 . . . . . 6  |-  (DECID  ph  ->  (DECID  ( ps  <->  ch )  ->  (
( ph  /\  -.  ( ps 
<->  ch ) )  <->  -.  ( ph  ->  ( ps  <->  ch )
) ) ) )
43imp 122 . . . . 5  |-  ( (DECID  ph  /\ DECID  ( ps  <->  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ph  ->  ( ps  <->  ch )
) ) )
5 pm5.32 441 . . . . . 6  |-  ( (
ph  ->  ( ps  <->  ch )
)  <->  ( ( ph  /\ 
ps )  <->  ( ph  /\ 
ch ) ) )
65notbii 629 . . . . 5  |-  ( -.  ( ph  ->  ( ps 
<->  ch ) )  <->  -.  (
( ph  /\  ps )  <->  (
ph  /\  ch )
) )
74, 6syl6bb 194 . . . 4  |-  ( (DECID  ph  /\ DECID  ( ps  <->  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) ) )
82, 7sylan2 280 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  -.  ( ps  <->  ch )
)  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\ 
ch ) ) ) )
9 xornbidc 1327 . . . . . 6  |-  (DECID  ps  ->  (DECID  ch 
->  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch ) ) ) )
109imp 122 . . . . 5  |-  ( (DECID  ps 
/\ DECID  ch )  ->  ( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch )
) )
1110adantl 271 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ps  \/_  ch )  <->  -.  ( ps  <->  ch ) ) )
1211anbi2d 452 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ph  /\  -.  ( ps  <->  ch )
) ) )
13 dcan 880 . . . . . 6  |-  (DECID  ph  ->  (DECID  ps 
-> DECID  ( ph  /\  ps )
) )
1413imp 122 . . . . 5  |-  ( (DECID  ph  /\ DECID  ps )  -> DECID 
( ph  /\  ps )
)
1514adantrr 463 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> DECID  ( ph  /\  ps ) )
16 dcan 880 . . . . . 6  |-  (DECID  ph  ->  (DECID  ch 
-> DECID  ( ph  /\  ch )
) )
1716imp 122 . . . . 5  |-  ( (DECID  ph  /\ DECID  ch )  -> DECID 
( ph  /\  ch )
)
1817adantrl 462 . . . 4  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> DECID  ( ph  /\  ch ) )
19 xornbidc 1327 . . . 4  |-  (DECID  ( ph  /\ 
ps )  ->  (DECID  ( ph  /\  ch )  -> 
( ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) )  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\  ch ) ) ) ) )
2015, 18, 19sylc 61 . . 3  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) )  <->  -.  ( ( ph  /\  ps )  <->  ( ph  /\  ch ) ) ) )
218, 12, 203bitr4d 218 . 2  |-  ( (DECID  ph  /\  (DECID  ps  /\ DECID  ch ) )  -> 
( ( ph  /\  ( ps  \/_  ch )
)  <->  ( ( ph  /\ 
ps )  \/_  ( ph  /\  ch ) ) ) )
2221exp32 357 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 102    <-> wb 103  DECID wdc 780    \/_ wxo 1311
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-xor 1312
This theorem is referenced by: (None)
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