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Theorem difundi 4274
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem difundi
StepHypRef Expression
1 dfun3 4260 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
21difeq2i 4114 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3 inindi 4221 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 4255 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5 invdif 4263 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 4263 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6ineq12i 4205 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴𝐶))
83, 4, 73eqtr3i 2762 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ (𝐴𝐶))
92, 8eqtri 2754 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3468  cdif 3940  cun 3941  cin 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950
This theorem is referenced by:  undm  4282  uncld  22895  inmbl  25421  difuncomp  32289  clsun  35720  poimirlem8  37008  ntrclskb  43378  ntrclsk3  43379  ntrclsk13  43380
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