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Theorem dn1 932
Description: A single axiom for Boolean algebra known as DN1. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jeffrey Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
Assertion
Ref Expression
dn1  |-  ( -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  ch )

Proof of Theorem dn1
StepHypRef Expression
1 pm2.45 386 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
2 imnan 411 . . . . 5  |-  ( ( -.  ( ph  \/  ps )  ->  -.  ph ) 
<->  -.  ( -.  ( ph  \/  ps )  /\  ph ) )
31, 2mpbi 199 . . . 4  |-  -.  ( -.  ( ph  \/  ps )  /\  ph )
43biorfi 396 . . 3  |-  ( ch  <->  ( ch  \/  ( -.  ( ph  \/  ps )  /\  ph ) ) )
5 orcom 376 . . . 4  |-  ( ( ch  \/  ( -.  ( ph  \/  ps )  /\  ph ) )  <-> 
( ( -.  ( ph  \/  ps )  /\  ph )  \/  ch )
)
6 ordir 835 . . . 4  |-  ( ( ( -.  ( ph  \/  ps )  /\  ph )  \/  ch )  <->  ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  ch ) ) )
75, 6bitri 240 . . 3  |-  ( ( ch  \/  ( -.  ( ph  \/  ps )  /\  ph ) )  <-> 
( ( -.  ( ph  \/  ps )  \/ 
ch )  /\  ( ph  \/  ch ) ) )
84, 7bitri 240 . 2  |-  ( ch  <->  ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  ch ) ) )
9 pm4.45 669 . . . . 5  |-  ( ch  <->  ( ch  /\  ( ch  \/  th ) ) )
10 anor 475 . . . . 5  |-  ( ( ch  /\  ( ch  \/  th ) )  <->  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )
119, 10bitri 240 . . . 4  |-  ( ch  <->  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) )
1211orbi2i 505 . . 3  |-  ( (
ph  \/  ch )  <->  (
ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )
1312anbi2i 675 . 2  |-  ( ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  ch ) )  <->  ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) )
14 anor 475 . 2  |-  ( ( ( -.  ( ph  \/  ps )  \/  ch )  /\  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) ) )
158, 13, 143bitrri 263 1  |-  ( -.  ( -.  ( -.  ( ph  \/  ps )  \/  ch )  \/  -.  ( ph  \/  -.  ( -.  ch  \/  -.  ( ch  \/  th ) ) ) )  <->  ch )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
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 177  df-or 359  df-an 360
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