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Theorem difindi 3839
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 3824 . . 3 (𝐵𝐶) = (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))
21difeq2i 3686 . 2 (𝐴 ∖ (𝐵𝐶)) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
3 indi 3831 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶)))
4 dfin2 3821 . . 3 (𝐴 ∩ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))) = (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶))))
5 invdif 3826 . . . 4 (𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
6 invdif 3826 . . . 4 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
75, 6uneq12i 3726 . . 3 ((𝐴 ∩ (V ∖ 𝐵)) ∪ (𝐴 ∩ (V ∖ 𝐶))) = ((𝐴𝐵) ∪ (𝐴𝐶))
83, 4, 73eqtr3i 2639 . 2 (𝐴 ∖ (V ∖ ((V ∖ 𝐵) ∪ (V ∖ 𝐶)))) = ((𝐴𝐵) ∪ (𝐴𝐶))
92, 8eqtri 2631 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3172  cdif 3536  cun 3537  cin 3538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546
This theorem is referenced by:  difdif2  3842  indm  3844  fndifnfp  6325  dprddisj2  18207  fctop  20560  cctop  20562  mretopd  20648  restcld  20728  cfinfil  21449  csdfil  21450  indifundif  28546  difres  28601  unelcarsg  29507  clsk3nimkb  37154  ntrclskb  37183  ntrclsk3  37184  ntrclsk13  37185  salincl  39016
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