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Theorem difin2 3443
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 3187 . . . . 5  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
21pm4.71d 615 . . . 4  |-  ( A 
C_  C  ->  (
x  e.  A  <->  ( x  e.  A  /\  x  e.  C ) ) )
32anbi1d 685 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  /\  -.  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B )
) )
4 eldif 3175 . . 3  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
5 elin 3371 . . . 4  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( x  e.  ( C  \  B
)  /\  x  e.  A ) )
6 eldif 3175 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
76anbi1i 676 . . . 4  |-  ( ( x  e.  ( C 
\  B )  /\  x  e.  A )  <->  ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A ) )
8 ancom 437 . . . . 5  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
9 anass 630 . . . . 5  |-  ( ( ( x  e.  A  /\  x  e.  C
)  /\  -.  x  e.  B )  <->  ( x  e.  A  /\  (
x  e.  C  /\  -.  x  e.  B
) ) )
108, 9bitr4i 243 . . . 4  |-  ( ( ( x  e.  C  /\  -.  x  e.  B
)  /\  x  e.  A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
115, 7, 103bitri 262 . . 3  |-  ( x  e.  ( ( C 
\  B )  i^i 
A )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  -.  x  e.  B
) )
123, 4, 113bitr4g 279 . 2  |-  ( A 
C_  C  ->  (
x  e.  ( A 
\  B )  <->  x  e.  ( ( C  \  B )  i^i  A
) ) )
1312eqrdv 2294 1  |-  ( A 
C_  C  ->  ( A  \  B )  =  ( ( C  \  B )  i^i  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164    C_ wss 3165
This theorem is referenced by:  issubdrg  15586  restcld  16919  limcnlp  19244  ballotlemfp1  23066  difelsiga  23509  difin2OLD  26464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179
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