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Theorem or3di 23904
Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
Assertion
Ref Expression
or3di  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta )
) )

Proof of Theorem or3di
StepHypRef Expression
1 df-3an 938 . . . 4  |-  ( ( ps  /\  ch  /\  ta )  <->  ( ( ps 
/\  ch )  /\  ta ) )
21orbi2i 506 . . 3  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ph  \/  ( ( ps  /\  ch )  /\  ta )
) )
3 ordi 835 . . 3  |-  ( (
ph  \/  ( ( ps  /\  ch )  /\  ta ) )  <->  ( ( ph  \/  ( ps  /\  ch ) )  /\  ( ph  \/  ta ) ) )
4 ordi 835 . . . 4  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )
54anbi1i 677 . . 3  |-  ( ( ( ph  \/  ( ps  /\  ch ) )  /\  ( ph  \/  ta ) )  <->  ( (
( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
62, 3, 53bitri 263 . 2  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ( (
ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
7 df-3an 938 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta ) )  <-> 
( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
86, 7bitr4i 244 1  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936
This theorem is referenced by:  or3dir  23905
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-or 360  df-an 361  df-3an 938
  Copyright terms: Public domain W3C validator