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

Theorem hadnot 1402
Description: The half adder distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadnot  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )

Proof of Theorem hadnot
StepHypRef Expression
1 xorneg 1322 . . . 4  |-  ( ( -.  ph  \/_  -.  ps ) 
<->  ( ph  \/_  ps ) )
2 biid 228 . . . 4  |-  ( -. 
ch 
<->  -.  ch )
31, 2xorbi12i 1323 . . 3  |-  ( ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) 
<->  ( ( ph  \/_  ps )  \/_  -.  ch )
)
4 xorneg2 1321 . . 3  |-  ( ( ( ph  \/_  ps )  \/_  -.  ch )  <->  -.  ( ( ph  \/_  ps )  \/_  ch ) )
53, 4bitr2i 242 . 2  |-  ( -.  ( ( ph  \/_  ps )  \/_  ch )  <->  ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) )
6 df-had 1389 . . 3  |-  (hadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/_ 
ps )  \/_  ch ) )
76notbii 288 . 2  |-  ( -. hadd
( ph ,  ps ,  ch )  <->  -.  ( ( ph  \/_  ps )  \/_  ch ) )
8 df-had 1389 . 2  |-  (hadd ( -.  ph ,  -.  ps ,  -.  ch )  <->  ( ( -.  ph  \/_  -.  ps )  \/_  -.  ch ) )
95, 7, 83bitr4i 269 1  |-  ( -. hadd
( ph ,  ps ,  ch )  <-> hadd ( -.  ph ,  -.  ps ,  -.  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/_ wxo 1313  haddwhad 1387
This theorem is referenced by:  had0  1412
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
  Copyright terms: Public domain W3C validator