Theorem difundir 4214

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 4209	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4202	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶)
3		invdif 4202	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
4		invdif 4202	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
5	3, 4	uneq12i 4095	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
6	1, 2, 5	3eqtr3i 2774	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894

This theorem is referenced by:  dfsymdif3  4230  difun2  4414  diftpsn3  4735  strleun  16858  setsfun0  16873  mreexmrid  17352  mreexexlem2d  17354  mvdco  19053  dprd2da  19645  dmdprdsplit2lem  19648  ablfac1eulem  19675  lbsextlem4  20423  opsrtoslem2  21263  nulmbl2  24700  uniioombllem3  24749  ex-dif  28787  indifundif  30873  imadifxp  30940  fzodif1  31114  cycpmrn  31410  ballotlemfp1  32458  ballotlemgun  32491  sltlpss  34087  onint1  34638  lindsadd  35770  lindsenlbs  35772  poimirlem2  35779  poimirlem6  35783  poimirlem7  35784  poimirlem8  35785  poimirlem22  35799  dvmptfprodlem  43485  fourierdlem102  43749  fourierdlem114  43761  caragenuncllem  44050  carageniuncllem1  44059