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Theorem difindi 3914
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 3899 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
21difeq2i 3758 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3 indi 3906 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 3893 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5 invdif 3901 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 3901 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6uneq12i 3798 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴𝐶))
83, 4, 73eqtr3i 2681 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴𝐵) ∪ (𝐴𝐶))
92, 8eqtri 2673 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  Vcvv 3231  cdif 3604  cun 3605  cin 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614
This theorem is referenced by:  difdif2  3917  indm  3919  fndifnfp  6483  dprddisj2  18484  fctop  20856  cctop  20858  mretopd  20944  restcld  21024  cfinfil  21744  csdfil  21745  indifundif  29482  difres  29539  unelcarsg  30502  clsk3nimkb  38655  ntrclskb  38684  ntrclsk3  38685  ntrclsk13  38686  salincl  40861
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