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Theorem difin2 3685
Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
difin2  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )

Proof of Theorem difin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3411 . . . . 5  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
21pm4.71d 632 . . . 4  |-  ( A 
C_  C  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  C ) ) )
32anbi1d 702 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B )
) )
4 eldif 3399 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3601 . . . 4  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( x  e.  ( C  \  B
)  /\  x  e.  A ) )
6 eldif 3399 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
76anbi1i 693 . . . 4  |-  ( ( x  e.  ( C 
\  B )  /\  x  e.  A )  <->  ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A ) )
8 ancom 448 . . . . 5  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
9 anass 647 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
108, 9bitr4i 252 . . . 4  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
115, 7, 103bitri 271 . . 3  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
123, 4, 113bitr4g 288 . 2  |-  ( A 
C_  C  ->  (
x  e.  ( A 
\  B )  <->  x  e.  ( ( C  \  B )  i^i  A
) ) )
1312eqrdv 2379 1  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826    \ cdif 3386    i^i cin 3388    C_ wss 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403
This theorem is referenced by:  gsumdifsnd  17101  issubdrg  17567  restcld  19759  limcnlp  22367  difelsiga  28282  sigapildsyslem  28306  difelcarsg2  28440  ballotlemfp1  28613  asindmre  30268  gsumdifsndf  33155
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