Theorem difundir 4238

Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difundir	⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

Proof of Theorem difundir

Step	Hyp	Ref	Expression
1		indir 4233	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4226	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶)
3		invdif 4226	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
4		invdif 4226	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
5	3, 4	uneq12i 4113	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
6	1, 2, 5	3eqtr3i 2762	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

Syntax hints:   = wceq 1541  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904

This theorem is referenced by:  dfsymdif3  4253  difun2  4428  diftpsn3  4751  strleun  17068  setsfun0  17083  mreexmrid  17549  mreexexlem2d  17551  chnccats1  18531  mvdco  19357  dprd2da  19956  dmdprdsplit2lem  19959  ablfac1eulem  19986  lbsextlem4  21098  opsrtoslem2  21991  nulmbl2  25464  uniioombllem3  25513  sltlpss  27853  slelss  27854  ex-dif  30403  indifundif  32504  imadifxp  32581  fzodif1  32775  cycpmrn  33112  ballotlemfp1  34505  ballotlemgun  34538  onint1  36493  lindsadd  37652  lindsenlbs  37654  poimirlem2  37661  poimirlem6  37665  poimirlem7  37666  poimirlem8  37667  poimirlem22  37681  dmxrnuncnvepres  38415  dvmptfprodlem  46041  fourierdlem102  46305  fourierdlem114  46317  caragenuncllem  46609  carageniuncllem1  46618