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Theorem difcom 3712
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 3491 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21sseq2i 3373 . 2  |-  ( A 
C_  ( B  u.  C )  <->  A  C_  ( C  u.  B )
)
3 ssundif 3711 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  \  B )  C_  C
)
4 ssundif 3711 . 2  |-  ( A 
C_  ( C  u.  B )  <->  ( A  \  C )  C_  B
)
52, 3, 43bitr3i 267 1  |-  ( ( A  \  B ) 
C_  C  <->  ( A  \  C )  C_  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \ cdif 3317    u. cun 3318    C_ wss 3320
This theorem is referenced by:  pssdifcom1  3713  pssdifcom2  3714  isreg2  17441  restmetu  18617  conss1  27623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334
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