Theorem difundir 4239

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 4234	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4227	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶)
3		invdif 4227	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐶)) = (𝐴 ∖ 𝐶)
4		invdif 4227	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
5	3, 4	uneq12i 4168	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐶)) ∪ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))
6	1, 2, 5	3eqtr3i 2794	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094  df-un 4154

This theorem is used by:  dfsymdif3  4254  difun2  4440  diftpsn3  4765  strleun  17361  setsfun0  17376  mreexmrid  17843  mreexexlem2d  17845  chnccats1  18825  mvdco  19686  dprd2da  20285  dmdprdsplit2lem  20288  ablfac1eulem  20315  lbsextlem4  21466  lindsenlbs  22184  opsrtoslem2  22392  nulmbl2  25884  uniioombllem3  25933  ltslpss  28321  leslss  28322  ex-dif  31051  indifundif  33189  imadifxp  33264  fzodif1  33451  cycpmrn  33771  ballotlemfp1  35191  ballotlemgun  35224  onint1  37414  lindsadd  38727  poimirlem2  38731  poimirlem6  38735  poimirlem7  38736  poimirlem8  38737  poimirlem22  38751  dmxrnuncnvepres  39515  dvmptfprodlem  47157  fourierdlem102  47421  fourierdlem114  47433  caragenuncllem  47725  carageniuncllem1  47734