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Theorem difundir 3856
Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundir ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem difundir
StepHypRef Expression
1 indir 3851 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2 invdif 3844 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴𝐵) ∖ 𝐶)
3 invdif 3844 . . 3 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
4 invdif 3844 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
53, 4uneq12i 3743 . 2 ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐶) ∪ (𝐵𝐶))
61, 2, 53eqtr3i 2651 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3186  cdif 3552  cun 3553  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562
This theorem is referenced by:  dfsymdif3  3869  difun2  4020  diftpsn3  4301  diftpsn3OLD  4302  setsfun0  15815  strleun  15893  mreexmrid  16224  mreexexlem2d  16226  mvdco  17786  dprd2da  18362  dmdprdsplit2lem  18365  ablfac1eulem  18392  lbsextlem4  19080  opsrtoslem2  19404  nulmbl2  23211  uniioombllem3  23259  ex-dif  27134  indifundif  29203  imadifxp  29259  ballotlemfp1  30334  ballotlemgun  30367  onint1  32090  lindsenlbs  33036  poimirlem2  33043  poimirlem6  33047  poimirlem7  33048  poimirlem8  33049  poimirlem22  33063  dvmptfprodlem  39465  fourierdlem102  39732  fourierdlem114  39744  caragenuncllem  40033  carageniuncllem1  40042
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