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Theorem difindi 4298
Description: Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem difindi
StepHypRef Expression
1 dfin3 4283 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
21difeq2i 4133 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3 indi 4290 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 4277 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5 invdif 4285 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 4285 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6uneq12i 4176 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴𝐶))
83, 4, 73eqtr3i 2771 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴𝐵) ∪ (𝐴𝐶))
92, 8eqtri 2763 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cdif 3960  cun 3961  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970
This theorem is referenced by:  difdif2  4302  indm  4304  fndifnfp  7196  dprddisj2  20074  fctop  23027  cctop  23029  mretopd  23116  restcld  23196  cfinfil  23917  csdfil  23918  indifundif  32552  difres  32620  unelcarsg  34294  clsk3nimkb  44030  ntrclskb  44059  ntrclsk3  44060  ntrclsk13  44061  salincl  46280  iscnrm3rlem1  48737
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