MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hadbi123d Structured version   Unicode version

Theorem hadbi123d 1392
Description: Equality theorem for half adder. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
hadbid.1  |-  ( ph  ->  ( ps  <->  ch )
)
hadbid.2  |-  ( ph  ->  ( th  <->  ta )
)
hadbid.3  |-  ( ph  ->  ( et  <->  ze )
)
Assertion
Ref Expression
hadbi123d  |-  ( ph  ->  (hadd ( ps ,  th ,  et )  <-> hadd ( ch ,  ta ,  ze ) ) )

Proof of Theorem hadbi123d
StepHypRef Expression
1 hadbid.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
2 hadbid.2 . . . 4  |-  ( ph  ->  ( th  <->  ta )
)
31, 2xorbi12d 1325 . . 3  |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  ta ) ) )
4 hadbid.3 . . 3  |-  ( ph  ->  ( et  <->  ze )
)
53, 4xorbi12d 1325 . 2  |-  ( ph  ->  ( ( ( ps 
\/_  th )  \/_  et ) 
<->  ( ( ch  \/_  ta )  \/_  ze )
) )
6 df-had 1390 . 2  |-  (hadd ( ps ,  th ,  et )  <->  ( ( ps 
\/_  th )  \/_  et ) )
7 df-had 1390 . 2  |-  (hadd ( ch ,  ta ,  ze )  <->  ( ( ch 
\/_  ta )  \/_  ze ) )
85, 6, 73bitr4g 281 1  |-  ( ph  ->  (hadd ( ps ,  th ,  et )  <-> hadd ( ch ,  ta ,  ze ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/_ wxo 1314  haddwhad 1388
This theorem is referenced by:  hadbi123i  1394  sadfval  12969  sadval  12973
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
  Copyright terms: Public domain W3C validator