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Theorem ordi 834
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 28-Nov-2013.)
Assertion
Ref Expression
ordi  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )

Proof of Theorem ordi
StepHypRef Expression
1 jcab 833 . 2  |-  ( ( -.  ph  ->  ( ps 
/\  ch ) )  <->  ( ( -.  ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
2 df-or 359 . 2  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( -.  ph 
->  ( ps  /\  ch ) ) )
3 df-or 359 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
4 df-or 359 . . 3  |-  ( (
ph  \/  ch )  <->  ( -.  ph  ->  ch )
)
53, 4anbi12i 678 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  <->  ( ( -.  ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
61, 2, 53bitr4i 268 1  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  ordir  835  orddi  839  pm5.63  890  pm4.43  893  cadan  1382  undi  3416  undif4  3511  elnn1uz2  10294  or3di  23123
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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