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Theorem difcom 3915
Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
Assertion
Ref Expression
difcom  |-  ( ( A  \  B ) 
C_  C  <->  ( A  \  C )  C_  B
)

Proof of Theorem difcom
StepHypRef Expression
1 uncom 3644 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21sseq2i 3524 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
3 ssundif 3914 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  B )  C_  C
)
4 ssundif 3914 . 2  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
52, 3, 43bitr3i 275 1  |-  ( ( A  \  B ) 
C_  C  <->  ( A  \  C )  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \ cdif 3468    u. cun 3469    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485
This theorem is referenced by:  pssdifcom1  3916  pssdifcom2  3917  isreg2  20005  restmetu  21216  conss1  31557  icccncfext  31893
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