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Theorem difundir 3190
 Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundir ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem difundir
StepHypRef Expression
1 indir 3186 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2 invdif 3179 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴𝐵) ∖ 𝐶)
3 invdif 3179 . . 3 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
4 invdif 3179 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
53, 4uneq12i 3095 . 2 ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐶) ∪ (𝐵𝐶))
61, 2, 53eqtr3i 2068 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1243  Vcvv 2557   ∖ cdif 2914   ∪ cun 2915   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924 This theorem is referenced by:  symdif1  3202  difun2  3302  diftpsn3  3505
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