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Theorem orddi 867
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 863 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ( ch  /\  th ) )  /\  ( ps  \/  ( ch  /\  th ) ) ) )
2 ordi 862 . . 3  |-  ( (
ph  \/  ( ch  /\ 
th ) )  <->  ( ( ph  \/  ch )  /\  ( ph  \/  th )
) )
3 ordi 862 . . 3  |-  ( ( ps  \/  ( ch 
/\  th ) )  <->  ( ( ps  \/  ch )  /\  ( ps  \/  th )
) )
42, 3anbi12i 695 . 2  |-  ( ( ( ph  \/  ( ch  /\  th ) )  /\  ( ps  \/  ( ch  /\  th )
) )  <->  ( (
( ph  \/  ch )  /\  ( ph  \/  th ) )  /\  (
( ps  \/  ch )  /\  ( ps  \/  th ) ) ) )
51, 4bitri 249 1  |-  ( ( ( ph  /\  ps )  \/  ( ch  /\ 
th ) )  <->  ( (
( ph  \/  ch )  /\  ( ph  \/  th ) )  /\  (
( ps  \/  ch )  /\  ( ps  \/  th ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    /\ wa 367
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
This theorem is referenced by:  prneimg  4197  wl-cases2-dnf  30213
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