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Theorem difindi 4234
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 4219 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
21difeq2i 4072 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3 indi 4226 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 4213 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5 invdif 4221 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 4221 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6uneq12i 4113 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴𝐶))
83, 4, 73eqtr3i 2851 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴𝐵) ∪ (𝐴𝐶))
92, 8eqtri 2843 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3473  cdif 3909  cun 3910  cin 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-rab 3134  df-v 3475  df-dif 3915  df-un 3917  df-in 3919
This theorem is referenced by:  difdif2  4237  indm  4239  fndifnfp  6912  dprddisj2  19137  fctop  21585  cctop  21587  mretopd  21673  restcld  21753  cfinfil  22474  csdfil  22475  indifundif  30269  difres  30334  unelcarsg  31575  clsk3nimkb  40512  ntrclskb  40541  ntrclsk3  40542  ntrclsk13  40543  salincl  42743
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