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Theorem hadcoma 1378
Description: Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadcoma  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ps ,  ph ,  ch ) )

Proof of Theorem hadcoma
StepHypRef Expression
1 xorcom 1298 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )
2 biid 227 . . 3  |-  ( ch  <->  ch )
31, 2xorbi12i 1305 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ( ps  \/_  ph )  \/_  ch ) )
4 df-had 1370 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
5 df-had 1370 . 2  |-  (hadd ( ps ,  ph ,  ch )  <->  ( ( ps 
\/_  ph )  \/_  ch ) )
63, 4, 53bitr4i 268 1  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ps ,  ph ,  ch ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/_ wxo 1295  haddwhad 1368
This theorem is referenced by:  hadrot  1380  sadcom  12670
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 177  df-xor 1296  df-had 1370
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