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Theorem andir 768
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 767 . 2  |-  ( ( ch  /\  ( ph  \/  ps ) )  <->  ( ( ch  /\  ph )  \/  ( ch  /\  ps ) ) )
2 ancom 262 . 2  |-  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ch  /\  ( ph  \/  ps ) ) )
3 ancom 262 . . 3  |-  ( (
ph  /\  ch )  <->  ( ch  /\  ph )
)
4 ancom 262 . . 3  |-  ( ( ps  /\  ch )  <->  ( ch  /\  ps )
)
53, 4orbi12i 716 . 2  |-  ( ( ( ph  /\  ch )  \/  ( ps  /\ 
ch ) )  <->  ( ( ch  /\  ph )  \/  ( ch  /\  ps ) ) )
61, 2, 53bitr4i 210 1  |-  ( ( ( ph  \/  ps )  /\  ch )  <->  ( ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 664
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 665
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  anddi  770  dcan  880  excxor  1314  xordc1  1329  sbequilem  1766  rexun  3178  rabun2  3276  reuun2  3280  xpundir  4483  coundi  4919  mptun  5130  tpostpos  6011  ltxr  9215
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