MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  had1 Unicode version

Theorem had1 1408
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 1393 . . 3  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  <->  ps )  <->  ch ) )
2 biass 349 . . 3  |-  ( ( ( ph  <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
31, 2bitri 241 . 2  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) )
4 id 20 . . . 4  |-  ( ph  ->  ph )
5 biidd 229 . . . 4  |-  ( ph  ->  ( ( ps  <->  ch )  <->  ( ps  <->  ch ) ) )
64, 52thd 232 . . 3  |-  ( ph  ->  ( ph  <->  ( ( ps 
<->  ch )  <->  ( ps  <->  ch ) ) ) )
7 biass 349 . . 3  |-  ( ( ( ph  <->  ( ps  <->  ch ) )  <->  ( ps  <->  ch ) )  <->  ( ph  <->  ( ( ps  <->  ch )  <->  ( ps  <->  ch ) ) ) )
86, 7sylibr 204 . 2  |-  ( ph  ->  ( ( ph  <->  ( ps  <->  ch ) )  <->  ( ps  <->  ch ) ) )
93, 8syl5bb 249 1  |-  ( ph  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ps  <->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177  haddwhad 1384
This theorem is referenced by:  had0  1409  sadadd2lem2  12925
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
  Copyright terms: Public domain W3C validator