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Theorem ordi 740
Description: Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
Assertion
Ref Expression
ordi  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )

Proof of Theorem ordi
StepHypRef Expression
1 simpl 106 . . . 4  |-  ( ( ps  /\  ch )  ->  ps )
21orim2i 688 . . 3  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  -> 
( ph  \/  ps ) )
3 simpr 107 . . . 4  |-  ( ( ps  /\  ch )  ->  ch )
43orim2i 688 . . 3  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  -> 
( ph  \/  ch ) )
52, 4jca 294 . 2  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  -> 
( ( ph  \/  ps )  /\  ( ph  \/  ch ) ) )
6 orc 643 . . . 4  |-  ( ph  ->  ( ph  \/  ( ps  /\  ch ) ) )
76adantl 266 . . 3  |-  ( ( ( ph  \/  ps )  /\  ph )  -> 
( ph  \/  ( ps  /\  ch ) ) )
86adantr 265 . . . 4  |-  ( (
ph  /\  ch )  ->  ( ph  \/  ( ps  /\  ch ) ) )
9 olc 642 . . . 4  |-  ( ( ps  /\  ch )  ->  ( ph  \/  ( ps  /\  ch ) ) )
108, 9jaoian 719 . . 3  |-  ( ( ( ph  \/  ps )  /\  ch )  -> 
( ph  \/  ( ps  /\  ch ) ) )
117, 10jaodan 721 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  ->  ( ph  \/  ( ps  /\  ch ) ) )
125, 11impbii 121 1  |-  ( (
ph  \/  ( ps  /\ 
ch ) )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    \/ wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ordir  741  orddi  744  pm5.63dc  864  pm4.43  867  orbididc  871  undi  3213  undif4  3312  elnn1uz2  8641
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