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Theorem cadcoma 1404
Description: Commutative law for adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
cadcoma  |-  (cadd (
ph ,  ps ,  ch )  <-> cadd ( ps ,  ph ,  ch ) )

Proof of Theorem cadcoma
StepHypRef Expression
1 ancom 438 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
2 xorcom 1316 . . . 4  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )
32anbi2i 676 . . 3  |-  ( ( ch  /\  ( ph  \/_ 
ps ) )  <->  ( ch  /\  ( ps  \/_  ph )
) )
41, 3orbi12i 508 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\  ( ph  \/_  ps ) ) )  <->  ( ( ps  /\  ph )  \/  ( ch  /\  ( ps  \/_  ph ) ) ) )
5 df-cad 1390 . 2  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ch  /\  ( ph  \/_  ps ) ) ) )
6 df-cad 1390 . 2  |-  (cadd ( ps ,  ph ,  ch )  <->  ( ( ps 
/\  ph )  \/  ( ch  /\  ( ps  \/_  ph ) ) ) )
74, 5, 63bitr4i 269 1  |-  (cadd (
ph ,  ps ,  ch )  <-> cadd ( ps ,  ph ,  ch ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    \/_ wxo 1313  caddwcad 1388
This theorem is referenced by:  cadrot  1406  sadcom  12975
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-xor 1314  df-cad 1390
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