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Theorem difdifdir 3914
Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
Assertion
Ref Expression
difdifdir  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  ( B  \  C ) )

Proof of Theorem difdifdir
StepHypRef Expression
1 dif32 3761 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3739 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2499 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3810 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2499 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3744 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3899 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3691 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2498 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3655 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2499 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2499 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddif 3636 . . . . 5  |-  ( _V 
\  ( _V  \  C ) )  =  C
1413uneq2i 3655 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( ( _V  \  B )  u.  C
)
15 indm 3757 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( _V  \  B
)  u.  ( _V 
\  ( _V  \  C ) ) )
16 invdif 3739 . . . . . 6  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1716difeq2i 3619 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1815, 17eqtr3i 2498 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( _V  \  ( B  \  C ) )
1914, 18eqtr3i 2498 . . 3  |-  ( ( _V  \  B )  u.  C )  =  ( _V  \  ( B  \  C ) )
2019ineq2i 3697 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) )
21 invdif 3739 . 2  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2212, 20, 213eqtri 2500 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by: (None)
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