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Theorem difdifdir 3554
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 3444 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3423 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2319 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3492 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2319 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3428 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3539 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3374 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2318 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3339 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2319 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2319 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddif 3321 . . . . 5  |-  ( _V 
\  ( _V  \  C ) )  =  C
1413uneq2i 3339 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( ( _V  \  B )  u.  C
)
15 indm 3440 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( _V  \  B
)  u.  ( _V 
\  ( _V  \  C ) ) )
16 invdif 3423 . . . . . 6  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1716difeq2i 3304 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1815, 17eqtr3i 2318 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( _V  \  ( B  \  C ) )
1914, 18eqtr3i 2318 . . 3  |-  ( ( _V  \  B )  u.  C )  =  ( _V  \  ( B  \  C ) )
2019ineq2i 3380 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) )
21 invdif 3423 . 2  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2212, 20, 213eqtri 2320 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469
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