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Theorem andir 766
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 765 . 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 714 . 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 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:  anddi  768  dcan  876  excxor  1310  xordc1  1325  sbequilem  1760  rexun  3153  rabun2  3250  reuun2  3254  xpundir  4423  coundi  4852  mptun  5060  tpostpos  5913  ltxr  8927
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