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Theorem indir 4040
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem indir
StepHypRef Expression
1 indi 4038 . 2 (𝐶 ∩ (𝐴𝐵)) = ((𝐶𝐴) ∪ (𝐶𝐵))
2 incom 3967 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 3967 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 incom 3967 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4uneq12i 3927 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐶𝐴) ∪ (𝐶𝐵))
61, 2, 53eqtr4i 2797 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  cin 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-in 3739
This theorem is referenced by:  difundir  4045  undisj1  4190  disjpr2  4404  resundir  5587  predun  5889  cdaassen  9257  fin23lem26  9400  fpwwe2lem13  9717  neitr  21264  fiuncmp  21487  connsuba  21503  trfil2  21970  tsmsres  22226  trust  22312  restmetu  22654  volun  23603  uniioombllem3  23643  itgsplitioo  23895  ppiprm  25168  chtprm  25170  chtdif  25175  ppidif  25180  carsgclctunlem1  30761  ballotlemfp1  30936  ballotlemgun  30969  mrsubvrs  31799  mthmpps  31859  fixun  32392  mbfposadd  33812  iunrelexp0  38601  31prm  42120
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