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Theorem hadcomb 1379
Description: Commutative law for triple XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadcomb  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ph ,  ch ,  ps ) )

Proof of Theorem hadcomb
StepHypRef Expression
1 biid 227 . . 3  |-  ( ph  <->  ph )
2 xorcom 1298 . . 3  |-  ( ( ps \/_ ch )  <->  ( ch \/_ ps )
)
31, 2xorbi12i 1305 . 2  |-  ( (
ph \/_ ( ps \/_ ch ) )  <->  ( ph \/_ ( ch \/_ ps ) ) )
4 hadass 1376 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ph \/_ ( ps \/_ ch )
) )
5 hadass 1376 . 2  |-  (hadd (
ph ,  ch ,  ps )  <->  ( ph \/_ ( ch \/_ ps )
) )
63, 4, 53bitr4i 268 1  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ph ,  ch ,  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   \/_wxo 1295  haddwhad 1368
This theorem is referenced by:  hadrot  1380
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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