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Theorem had0 1412
Description: If the first parameter is false, the half adder is equivalent to the XOR of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
had0  |-  ( -. 
ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  \/_  ch )
) )

Proof of Theorem had0
StepHypRef Expression
1 had1 1411 . . 3  |-  ( -. 
ph  ->  (hadd ( -. 
ph ,  -.  ps ,  -.  ch )  <->  ( -.  ps 
<->  -.  ch ) ) )
2 hadnot 1402 . . 3  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
3 df-xor 1314 . . . . 5  |-  ( ( -.  ps  \/_  -.  ch )  <->  -.  ( -.  ps 
<->  -.  ch ) )
4 xorneg 1322 . . . . 5  |-  ( ( -.  ps  \/_  -.  ch )  <->  ( ps  \/_  ch ) )
53, 4bitr3i 243 . . . 4  |-  ( -.  ( -.  ps  <->  -.  ch )  <->  ( ps  \/_  ch )
)
65con1bii 322 . . 3  |-  ( -.  ( ps  \/_  ch ) 
<->  ( -.  ps  <->  -.  ch )
)
71, 2, 63bitr4g 280 . 2  |-  ( -. 
ph  ->  ( -. hadd ( ph ,  ps ,  ch ) 
<->  -.  ( ps  \/_  ch ) ) )
87con4bid 285 1  |-  ( -. 
ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  \/_  ch )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/_ wxo 1313  haddwhad 1387
This theorem is referenced by:  sadadd2lem2  12954  saddisjlem  12968
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 1314  df-had 1389
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