Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rp-fakeoranass Structured version   Visualization version   Unicode version

Theorem rp-fakeoranass 36229
Description: A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)
Assertion
Ref Expression
rp-fakeoranass  |-  ( (
ph  ->  ch )  <->  ( (
( ph  \/  ps )  /\  ch )  <->  ( ph  \/  ( ps  /\  ch ) ) ) )

Proof of Theorem rp-fakeoranass
StepHypRef Expression
1 rp-fakeanorass 36228 . 2  |-  ( (
ph  ->  ch )  <->  ( (
( ch  /\  ps )  \/  ph )  <->  ( ch  /\  ( ps  \/  ph ) ) ) )
2 bicom 205 . . 3  |-  ( ( ( ( ch  /\  ps )  \/  ph )  <->  ( ch  /\  ( ps  \/  ph ) ) )  <->  ( ( ch 
/\  ( ps  \/  ph ) )  <->  ( ( ch  /\  ps )  \/ 
ph ) ) )
3 ancom 457 . . . . 5  |-  ( ( ch  /\  ( ps  \/  ph ) )  <-> 
( ( ps  \/  ph )  /\  ch )
)
4 orcom 394 . . . . . 6  |-  ( ( ps  \/  ph )  <->  (
ph  \/  ps )
)
54anbi1i 709 . . . . 5  |-  ( ( ( ps  \/  ph )  /\  ch )  <->  ( ( ph  \/  ps )  /\  ch ) )
63, 5bitri 257 . . . 4  |-  ( ( ch  /\  ( ps  \/  ph ) )  <-> 
( ( ph  \/  ps )  /\  ch )
)
7 orcom 394 . . . . 5  |-  ( ( ( ch  /\  ps )  \/  ph )  <->  ( ph  \/  ( ch  /\  ps ) ) )
8 ancom 457 . . . . . 6  |-  ( ( ch  /\  ps )  <->  ( ps  /\  ch )
)
98orbi2i 528 . . . . 5  |-  ( (
ph  \/  ( ch  /\ 
ps ) )  <->  ( ph  \/  ( ps  /\  ch ) ) )
107, 9bitri 257 . . . 4  |-  ( ( ( ch  /\  ps )  \/  ph )  <->  ( ph  \/  ( ps  /\  ch ) ) )
116, 10bibi12i 322 . . 3  |-  ( ( ( ch  /\  ( ps  \/  ph ) )  <-> 
( ( ch  /\  ps )  \/  ph )
)  <->  ( ( (
ph  \/  ps )  /\  ch )  <->  ( ph  \/  ( ps  /\  ch ) ) ) )
122, 11bitri 257 . 2  |-  ( ( ( ( ch  /\  ps )  \/  ph )  <->  ( ch  /\  ( ps  \/  ph ) ) )  <->  ( ( (
ph  \/  ps )  /\  ch )  <->  ( ph  \/  ( ps  /\  ch ) ) ) )
131, 12bitri 257 1  |-  ( (
ph  ->  ch )  <->  ( (
( ph  \/  ps )  /\  ch )  <->  ( ph  \/  ( ps  /\  ch ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376
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 190  df-or 377  df-an 378
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator