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Theorem cadcomb 1405
Description: Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
cadcomb  |-  (cadd (
ph ,  ps ,  ch )  <-> cadd ( ph ,  ch ,  ps ) )

Proof of Theorem cadcomb
StepHypRef Expression
1 3orcoma 944 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch )  \/  ( ps  /\  ch ) )  <-> 
( ( ph  /\  ch )  \/  ( ph  /\  ps )  \/  ( ps  /\  ch ) ) )
2 biid 228 . . . 4  |-  ( (
ph  /\  ch )  <->  (
ph  /\  ch )
)
3 biid 228 . . . 4  |-  ( (
ph  /\  ps )  <->  (
ph  /\  ps )
)
4 ancom 438 . . . 4  |-  ( ( ps  /\  ch )  <->  ( ch  /\  ps )
)
52, 3, 43orbi123i 1143 . . 3  |-  ( ( ( ph  /\  ch )  \/  ( ph  /\ 
ps )  \/  ( ps  /\  ch ) )  <-> 
( ( ph  /\  ch )  \/  ( ph  /\  ps )  \/  ( ch  /\  ps ) ) )
61, 5bitri 241 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch )  \/  ( ps  /\  ch ) )  <-> 
( ( ph  /\  ch )  \/  ( ph  /\  ps )  \/  ( ch  /\  ps ) ) )
7 cador 1400 . 2  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
8 cador 1400 . 2  |-  (cadd (
ph ,  ch ,  ps )  <->  ( ( ph  /\ 
ch )  \/  ( ph  /\  ps )  \/  ( ch  /\  ps ) ) )
96, 7, 83bitr4i 269 1  |-  (cadd (
ph ,  ps ,  ch )  <-> cadd ( ph ,  ch ,  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    \/ w3o 935  caddwcad 1388
This theorem is referenced by:  cadrot  1406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-xor 1314  df-cad 1390
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