MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3bior2fd Structured version   Unicode version

Theorem 3bior2fd 1336
Description: A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 405. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
Hypotheses
Ref Expression
3biorfd.1  |-  ( ph  ->  -.  th )
3biorfd.2  |-  ( ph  ->  -.  ch )
Assertion
Ref Expression
3bior2fd  |-  ( ph  ->  ( ps  <->  ( th  \/  ch  \/  ps )
) )

Proof of Theorem 3bior2fd
StepHypRef Expression
1 3biorfd.2 . . 3  |-  ( ph  ->  -.  ch )
2 biorf 405 . . 3  |-  ( -. 
ch  ->  ( ps  <->  ( ch  \/  ps ) ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( ps  <->  ( ch  \/  ps ) ) )
4 3biorfd.1 . . 3  |-  ( ph  ->  -.  th )
543bior1fd 1334 . 2  |-  ( ph  ->  ( ( ch  \/  ps )  <->  ( th  \/  ch  \/  ps ) ) )
63, 5bitrd 253 1  |-  ( ph  ->  ( ps  <->  ( th  \/  ch  \/  ps )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    \/ w3o 972
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 185  df-or 370  df-3or 974
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator