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Theorem indifdir 3606
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir  |-  ( ( A  \  B )  i^i  C )  =  ( ( A  i^i  C )  \  ( B  i^i  C ) )

Proof of Theorem indifdir
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm3.24 877 . . . . . . . 8  |-  -.  (
x  e.  C  /\  -.  x  e.  C
)
21intnan 905 . . . . . . 7  |-  -.  (
x  e.  A  /\  ( x  e.  C  /\  -.  x  e.  C
) )
3 anass 649 . . . . . . 7  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  C
) ) )
42, 3mtbir 299 . . . . . 6  |-  -.  (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C )
54biorfi 407 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  <->  ( (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  \/  (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C ) ) )
6 an32 796 . . . . 5  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
7 andi 862 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  ( -.  x  e.  B  \/  -.  x  e.  C
) )  <->  ( (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  \/  (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C ) ) )
85, 6, 73bitr4i 277 . . . 4  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) ) )
9 ianor 488 . . . . 5  |-  ( -.  ( x  e.  B  /\  x  e.  C
)  <->  ( -.  x  e.  B  \/  -.  x  e.  C )
)
109anbi2i 694 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  (
x  e.  B  /\  x  e.  C )
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  ( -.  x  e.  B  \/  -.  x  e.  C
) ) )
118, 10bitr4i 252 . . 3  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
12 elin 3539 . . . 4  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
13 eldif 3338 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1413anbi1i 695 . . . 4  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
1512, 14bitri 249 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
16 eldif 3338 . . . 4  |-  ( x  e.  ( ( A  i^i  C )  \ 
( B  i^i  C
) )  <->  ( x  e.  ( A  i^i  C
)  /\  -.  x  e.  ( B  i^i  C
) ) )
17 elin 3539 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
18 elin 3539 . . . . . 6  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
1918notbii 296 . . . . 5  |-  ( -.  x  e.  ( B  i^i  C )  <->  -.  (
x  e.  B  /\  x  e.  C )
)
2017, 19anbi12i 697 . . . 4  |-  ( ( x  e.  ( A  i^i  C )  /\  -.  x  e.  ( B  i^i  C ) )  <-> 
( ( x  e.  A  /\  x  e.  C )  /\  -.  ( x  e.  B  /\  x  e.  C
) ) )
2116, 20bitri 249 . . 3  |-  ( x  e.  ( ( A  i^i  C )  \ 
( B  i^i  C
) )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
2211, 15, 213bitr4i 277 . 2  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  x  e.  ( ( A  i^i  C )  \  ( B  i^i  C ) ) )
2322eqriv 2440 1  |-  ( ( A  \  B )  i^i  C )  =  ( ( A  i^i  C )  \  ( B  i^i  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3325    i^i cin 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-dif 3331  df-in 3335
This theorem is referenced by:  fresaun  5582  uniioombllem4  21066  preddif  27652
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