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Theorem difin2 3605
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 3344 . . . . 5  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
21pm4.71d 617 . . . 4  |-  ( A 
C_  C  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  C ) ) )
32anbi1d 687 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B )
) )
4 eldif 3332 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3532 . . . 4  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( x  e.  ( C  \  B
)  /\  x  e.  A ) )
6 eldif 3332 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
76anbi1i 678 . . . 4  |-  ( ( x  e.  ( C 
\  B )  /\  x  e.  A )  <->  ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A ) )
8 ancom 439 . . . . 5  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
9 anass 632 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
108, 9bitr4i 245 . . . 4  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
115, 7, 103bitri 264 . . 3  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
123, 4, 113bitr4g 281 . 2  |-  ( A 
C_  C  ->  (
x  e.  ( A 
\  B )  <->  x  e.  ( ( C  \  B )  i^i  A
) ) )
1312eqrdv 2436 1  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322
This theorem is referenced by:  issubdrg  15895  restcld  17238  limcnlp  19767  difelsiga  24518  ballotlemfp1  24751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336
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