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Theorem difcom 3880
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 3610 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21sseq2i 3489 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
3 ssundif 3879 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  B )  C_  C
)
4 ssundif 3879 . 2  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
52, 3, 43bitr3i 278 1  |-  ( ( A  \  B ) 
C_  C  <->  ( A  \  C )  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \ cdif 3433    u. cun 3434    C_ wss 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450
This theorem is referenced by:  pssdifcom1  3881  pssdifcom2  3882  isreg2  20379  restmetu  21571  conss1  36654  icccncfext  37584
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