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Theorem indifdir 3589
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 pm3.24 853 . . . . . . . 8  |-  -.  (
x  e.  C  /\  -.  x  e.  C
)
21intnan 881 . . . . . . 7  |-  -.  (
x  e.  A  /\  ( x  e.  C  /\  -.  x  e.  C
) )
3 anass 631 . . . . . . 7  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  C
) ) )
42, 3mtbir 291 . . . . . 6  |-  -.  (
( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  C )
54biorfi 397 . . . . 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 774 . . . . 5  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
7 andi 838 . . . . 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 269 . . . 4  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( -.  x  e.  B  \/  -.  x  e.  C ) ) )
9 ianor 475 . . . . 5  |-  ( -.  ( x  e.  B  /\  x  e.  C
)  <->  ( -.  x  e.  B  \/  -.  x  e.  C )
)
109anbi2i 676 . . . 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 244 . . 3  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  ( x  e.  B  /\  x  e.  C ) ) )
12 elin 3522 . . . 4  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( x  e.  ( A  \  B
)  /\  x  e.  C ) )
13 eldif 3322 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1413anbi1i 677 . . . 4  |-  ( ( x  e.  ( A 
\  B )  /\  x  e.  C )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
1512, 14bitri 241 . . 3  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  /\  x  e.  C ) )
16 eldif 3322 . . . 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 3522 . . . . 5  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
18 elin 3522 . . . . . 6  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
1918notbii 288 . . . . 5  |-  ( -.  x  e.  ( B  i^i  C )  <->  -.  (
x  e.  B  /\  x  e.  C )
)
2017, 19anbi12i 679 . . . 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 241 . . 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 269 . 2  |-  ( x  e.  ( ( A 
\  B )  i^i 
C )  <->  x  e.  ( ( A  i^i  C )  \  ( B  i^i  C ) ) )
2322eqriv 2432 1  |-  ( ( A  \  B )  i^i  C )  =  ( ( A  i^i  C )  \  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309    i^i cin 3311
This theorem is referenced by:  fresaun  5605  uniioombllem4  19466  preddif  25446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319
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