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Theorem difdifdir 3828
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 3679 . . . . 5  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
2 invdif 3657 . . . . 5  |-  ( ( A  \  C )  i^i  ( _V  \  B ) )  =  ( ( A  \  C )  \  B
)
31, 2eqtr4i 2453 . . . 4  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  ( _V  \  B ) )
4 un0 3732 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( A  \  C
)  i^i  ( _V  \  B ) )
53, 4eqtr4i 2453 . . 3  |-  ( ( A  \  B ) 
\  C )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
6 indi 3662 . . . 4  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (
( A  \  C
)  i^i  C )
)
7 disjdif 3812 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  (/)
8 incom 3598 . . . . . 6  |-  ( C  i^i  ( A  \  C ) )  =  ( ( A  \  C )  i^i  C
)
97, 8eqtr3i 2452 . . . . 5  |-  (/)  =  ( ( A  \  C
)  i^i  C )
109uneq2i 3560 . . . 4  |-  ( ( ( A  \  C
)  i^i  ( _V  \  B ) )  u.  (/) )  =  (
( ( A  \  C )  i^i  ( _V  \  B ) )  u.  ( ( A 
\  C )  i^i 
C ) )
116, 10eqtr4i 2453 . . 3  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( ( A 
\  C )  i^i  ( _V  \  B
) )  u.  (/) )
125, 11eqtr4i 2453 . 2  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  i^i  (
( _V  \  B
)  u.  C ) )
13 ddif 3540 . . . . 5  |-  ( _V 
\  ( _V  \  C ) )  =  C
1413uneq2i 3560 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( ( _V  \  B )  u.  C
)
15 indm 3675 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( ( _V  \  B
)  u.  ( _V 
\  ( _V  \  C ) ) )
16 invdif 3657 . . . . . 6  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
1716difeq2i 3523 . . . . 5  |-  ( _V 
\  ( B  i^i  ( _V  \  C ) ) )  =  ( _V  \  ( B 
\  C ) )
1815, 17eqtr3i 2452 . . . 4  |-  ( ( _V  \  B )  u.  ( _V  \ 
( _V  \  C
) ) )  =  ( _V  \  ( B  \  C ) )
1914, 18eqtr3i 2452 . . 3  |-  ( ( _V  \  B )  u.  C )  =  ( _V  \  ( B  \  C ) )
2019ineq2i 3604 . 2  |-  ( ( A  \  C )  i^i  ( ( _V 
\  B )  u.  C ) )  =  ( ( A  \  C )  i^i  ( _V  \  ( B  \  C ) ) )
21 invdif 3657 . 2  |-  ( ( A  \  C )  i^i  ( _V  \ 
( B  \  C
) ) )  =  ( ( A  \  C )  \  ( B  \  C ) )
2212, 20, 213eqtri 2454 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  ( B  \  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3022    \ cdif 3376    u. cun 3377    i^i cin 3378   (/)c0 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705
This theorem is referenced by: (None)
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