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Theorem ordi 837
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 836 . 2  |-  ( ( -.  ph  ->  ( ps 
/\  ch ) )  <->  ( ( -.  ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
2 df-or 361 . 2  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( -.  ph 
->  ( ps  /\  ch ) ) )
3 df-or 361 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
4 df-or 361 . . 3  |-  ( (
ph  \/  ch )  <->  ( -.  ph  ->  ch )
)
53, 4anbi12i 681 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  <->  ( ( -.  ph  ->  ps )  /\  ( -.  ph  ->  ch ) ) )
61, 2, 53bitr4i 270 1  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360
This theorem is referenced by:  ordir  838  jcabOLD  839  orddi  844  pm5.63  895  pm4.43  898  cadan  1388  undi  3358  undif4  3453  elnn1uz2  10226
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362
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