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Theorem difundi 3838
 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 3824 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))
21difeq2i 3687 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
3 inindi 3792 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 3822 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))))
5 invdif 3827 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 3827 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6ineq12i 3774 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∩ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∩ (𝐴𝐶))
83, 4, 73eqtr3i 2640 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∩ (V ∖ 𝐶)))) = ((𝐴𝐵) ∩ (𝐴𝐶))
92, 8eqtri 2632 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547 This theorem is referenced by:  undm  3844  uncld  20655  inmbl  23117  difuncomp  28752  clsun  31493  poimirlem8  32587  ntrclskb  37387  ntrclsk3  37388  ntrclsk13  37389
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