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Theorem hadcoma 1495
Description: Commutative law for the adder sum. (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 1403 . . 3  |-  ( (
ph  \/_  ps )  <->  ( ps  \/_  ph ) )
2 biid 239 . . 3  |-  ( ch  <->  ch )
31, 2xorbi12i 1416 . 2  |-  ( ( ( ph  \/_  ps )  \/_  ch )  <->  ( ( ps  \/_  ph )  \/_  ch ) )
4 df-had 1490 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
5 df-had 1490 . 2  |-  (hadd ( ps ,  ph ,  ch )  <->  ( ( ps 
\/_  ph )  \/_  ch ) )
63, 4, 53bitr4i 280 1  |-  (hadd (
ph ,  ps ,  ch )  <-> hadd ( ps ,  ph ,  ch ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/_ wxo 1400  haddwhad 1489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 188  df-xor 1401  df-had 1490
This theorem is referenced by:  hadrot  1497  sadcom  14436
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