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Theorem or3di 27568
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 973 . . . 4  |-  ( ( ps  /\  ch  /\  ta )  <->  ( ( ps 
/\  ch )  /\  ta ) )
21orbi2i 517 . . 3  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ph  \/  ( ( ps  /\  ch )  /\  ta )
) )
3 ordi 862 . . 3  |-  ( (
ph  \/  ( ( ps  /\  ch )  /\  ta ) )  <->  ( ( ph  \/  ( ps  /\  ch ) )  /\  ( ph  \/  ta ) ) )
4 ordi 862 . . . 4  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )
54anbi1i 693 . . 3  |-  ( ( ( ph  \/  ( ps  /\  ch ) )  /\  ( ph  \/  ta ) )  <->  ( (
( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
62, 3, 53bitri 271 . 2  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ( (
ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
7 df-3an 973 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta ) )  <-> 
( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ph  \/  ta ) ) )
86, 7bitr4i 252 1  |-  ( (
ph  \/  ( ps  /\ 
ch  /\  ta )
)  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ph  \/  ta )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971
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 368  df-an 369  df-3an 973
This theorem is referenced by:  or3dir  27569
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