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Theorem difin2 3710
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 3448 . . . . 5  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
21pm4.71d 634 . . . 4  |-  ( A 
C_  C  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  C ) ) )
32anbi1d 704 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B )
) )
4 eldif 3436 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3637 . . . 4  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( x  e.  ( C  \  B
)  /\  x  e.  A ) )
6 eldif 3436 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
76anbi1i 695 . . . 4  |-  ( ( x  e.  ( C 
\  B )  /\  x  e.  A )  <->  ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A ) )
8 ancom 450 . . . . 5  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
9 anass 649 . . . . 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 2448 1  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3423    i^i cin 3425    C_ wss 3426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3070  df-dif 3429  df-in 3433  df-ss 3440
This theorem is referenced by:  issubdrg  16996  restcld  18892  limcnlp  21469  difelsiga  26710  ballotlemfp1  27008  asindmre  28617  gsumdifsnd  30900  gsumdifsndf  30903
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