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Theorem had1 1412
Description: If the first parameter is true, the half adder is equivalent to the equality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
had1  |-  ( ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  <->  ch ) ) )

Proof of Theorem had1
StepHypRef Expression
1 hadbi 1397 . . 3  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
2 biass 350 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
31, 2bitri 242 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
4 id 21 . . . 4  |-  ( ph  ->  ph )
5 biidd 230 . . . 4  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ps  <->  ch ) ) )
64, 52thd 233 . . 3  |-  ( ph  ->  ( ph  <->  ( ( ps 
<->  ch )  <->  ( ps  <->  ch ) ) ) )
7 biass 350 . . 3  |-  ( ( ( ph  <->  ( ps  <->  ch ) )  <->  ( ps  <->  ch ) )  <->  ( ph  <->  ( ( ps  <->  ch )  <->  ( ps  <->  ch ) ) ) )
86, 7sylibr 205 . 2  |-  ( ph  ->  ( ( ph  <->  ( ps  <->  ch ) )  <->  ( ps  <->  ch ) ) )
93, 8syl5bb 250 1  |-  ( ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  <->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178  haddwhad 1388
This theorem is referenced by:  had0  1413  sadadd2lem2  12967
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 179  df-xor 1315  df-had 1390
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