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Theorem andir 791
Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
Assertion
Ref Expression
andir  |-  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )

Proof of Theorem andir
StepHypRef Expression
1 andi 790 . 2  |-  ( ( ch  /\  ( ph  \/  ps ) )  <->  ( ( ch  /\  ph )  \/  ( ch  /\  ps ) ) )
2 ancom 264 . 2  |-  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ch  /\  ( ph  \/  ps ) ) )
3 ancom 264 . . 3  |-  ( (
ph  /\  ch )  <->  ( ch  /\  ph )
)
4 ancom 264 . . 3  |-  ( ( ps  /\  ch )  <->  ( ch  /\  ps )
)
53, 4orbi12i 736 . 2  |-  ( ( ( ph  /\  ch )  \/  ( ps  /\ 
ch ) )  <->  ( ( ch  /\  ph )  \/  ( ch  /\  ps ) ) )
61, 2, 53bitr4i 211 1  |-  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anddi  793  dcan  901  excxor  1339  xordc1  1354  sbequilem  1792  rexun  3224  rabun2  3323  reuun2  3327  xpundir  4564  coundi  5008  mptun  5222  tpostpos  6127  ltxr  9513
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