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Theorem hadcomb 1395
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 228 . . 3  |-  ( ph  <->  ph )
2 xorcom 1313 . . 3  |-  ( ( ps  \/_  ch )  <->  ( ch  \/_  ps )
)
31, 2xorbi12i 1320 . 2  |-  ( (
ph  \/_  ( ps  \/_ 
ch ) )  <->  ( ph  \/_  ( ch  \/_  ps ) ) )
4 hadass 1392 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ph  \/_  ( ps  \/_  ch ) ) )
5 hadass 1392 . 2  |-  (hadd (
ph ,  ch ,  ps )  <->  ( ph  \/_  ( ch  \/_  ps ) ) )
63, 4, 53bitr4i 269 1  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ph ,  ch ,  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/_ wxo 1310  haddwhad 1384
This theorem is referenced by:  hadrot  1396
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-xor 1311  df-had 1386
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