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Theorem difcom 3551
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 3332 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21sseq2i 3216 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
3 ssundif 3550 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  B )  C_  C
)
4 ssundif 3550 . 2  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
52, 3, 43bitr3i 266 1  |-  ( ( A  \  B ) 
C_  C  <->  ( A  \  C )  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \ cdif 3162    u. cun 3163    C_ wss 3165
This theorem is referenced by:  pssdifcom1  3552  pssdifcom2  3553  isreg2  17121  conss1  27750
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-un 3170  df-in 3172  df-ss 3179
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