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Theorem difundir 3736
Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundir  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )

Proof of Theorem difundir
StepHypRef Expression
1 indir 3731 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  i^i  ( _V  \  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )
2 invdif 3724 . 2  |-  ( ( A  u.  B )  i^i  ( _V  \  C ) )  =  ( ( A  u.  B )  \  C
)
3 invdif 3724 . . 3  |-  ( A  i^i  ( _V  \  C ) )  =  ( A  \  C
)
4 invdif 3724 . . 3  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
53, 4uneq12i 3641 . 2  |-  ( ( A  i^i  ( _V 
\  C ) )  u.  ( B  i^i  ( _V  \  C ) ) )  =  ( ( A  \  C
)  u.  ( B 
\  C ) )
61, 2, 53eqtr3i 2480 1  |-  ( ( A  u.  B ) 
\  C )  =  ( ( A  \  C )  u.  ( B  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   _Vcvv 3095    \ cdif 3458    u. cun 3459    i^i cin 3460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468
This theorem is referenced by:  symdif1  3748  difun2  3893  diftpsn3  4153  strleun  14708  mreexmrid  15021  mreexexlem2d  15023  mvdco  16448  dprd2da  17069  dmdprdsplit2lem  17072  ablfac1eulem  17101  lbsextlem4  17785  opsrtoslem2  18127  nulmbl2  21924  uniioombllem3  21971  ex-dif  25120  imadifxp  27434  ballotlemfp1  28407  ballotlemgun  28440  onint1  29889  fourierdlem102  31880  fourierdlem114  31892
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