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

Theorem consensus 930
Description: The consensus theorem. This theorem and its dual (with  \/ and  /\ interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term  ( ps  /\  ch ) on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
Assertion
Ref Expression
consensus  |-  ( ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  \/  ( ps 
/\  ch ) )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) )

Proof of Theorem consensus
StepHypRef Expression
1 id 21 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
2 orc 376 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
32adantrr 700 . . . 4  |-  ( (
ph  /\  ( ps  /\ 
ch ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
4 olc 375 . . . . 5  |-  ( ( -.  ph  /\  ch )  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
54adantrl 699 . . . 4  |-  ( ( -.  ph  /\  ( ps  /\  ch ) )  ->  ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ch )
) )
63, 5pm2.61ian 768 . . 3  |-  ( ( ps  /\  ch )  ->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
71, 6jaoi 370 . 2  |-  ( ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  \/  ( ps 
/\  ch ) )  -> 
( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
) )
8 orc 376 . 2  |-  ( ( ( ph  /\  ps )  \/  ( -.  ph 
/\  ch ) )  -> 
( ( ( ph  /\ 
ps )  \/  ( -.  ph  /\  ch )
)  \/  ( ps 
/\  ch ) ) )
97, 8impbii 182 1  |-  ( ( ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch )
)  \/  ( ps 
/\  ch ) )  <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    \/ wo 359    /\ wa 360
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
  Copyright terms: Public domain W3C validator