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Theorem orddi 767
Description: Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
orddi  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( (
( ph  \/  ch )  /\  ( ph  \/  th ) )  /\  (
( ps  \/  ch )  /\  ( ps  \/  th ) ) ) )

Proof of Theorem orddi
StepHypRef Expression
1 ordir 764 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ( ch  /\  th ) )  /\  ( ps  \/  ( ch  /\  th ) ) ) )
2 ordi 763 . . 3  |-  ( (
ph  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ch )  /\  ( ph  \/  th )
) )
3 ordi 763 . . 3  |-  ( ( ps  \/  ( ch 
/\  th ) )  <->  ( ( ps  \/  ch )  /\  ( ps  \/  th )
) )
42, 3anbi12i 448 . 2  |-  ( ( ( ph  \/  ( ch  /\  th ) )  /\  ( ps  \/  ( ch  /\  th )
) )  <->  ( (
( ph  \/  ch )  /\  ( ph  \/  th ) )  /\  (
( ps  \/  ch )  /\  ( ps  \/  th ) ) ) )
51, 4bitri 182 1  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( (
( ph  \/  ch )  /\  ( ph  \/  th ) )  /\  (
( ps  \/  ch )  /\  ( ps  \/  th ) ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  prneimg  3586
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