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Theorem anddi 842
Description: Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
anddi  |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  th ) )  <->  ( (
( ph  /\  ch )  \/  ( ph  /\  th ) )  \/  (
( ps  /\  ch )  \/  ( ps  /\ 
th ) ) ) )

Proof of Theorem anddi
StepHypRef Expression
1 andir 840 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  th ) )  <->  ( ( ph  /\  ( ch  \/  th ) )  \/  ( ps  /\  ( ch  \/  th ) ) ) )
2 andi 839 . . 3  |-  ( (
ph  /\  ( ch  \/  th ) )  <->  ( ( ph  /\  ch )  \/  ( ph  /\  th ) ) )
3 andi 839 . . 3  |-  ( ( ps  /\  ( ch  \/  th ) )  <-> 
( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
42, 3orbi12i 509 . 2  |-  ( ( ( ph  /\  ( ch  \/  th ) )  \/  ( ps  /\  ( ch  \/  th )
) )  <->  ( (
( ph  /\  ch )  \/  ( ph  /\  th ) )  \/  (
( ps  /\  ch )  \/  ( ps  /\ 
th ) ) ) )
51, 4bitri 242 1  |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  th ) )  <->  ( (
( ph  /\  ch )  \/  ( ph  /\  th ) )  \/  (
( ps  /\  ch )  \/  ( ps  /\ 
th ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360
This theorem is referenced by:  prnebg  3980  funun  5496  disjxpin  24029  undif3VD  28995
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 179  df-or 361  df-an 362
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