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Theorem dif32 3761
Description: Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
Assertion
Ref Expression
dif32  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)

Proof of Theorem dif32
StepHypRef Expression
1 uncom 3648 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21difeq2i 3619 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( C  u.  B )
)
3 difun1 3758 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
4 difun1 3758 . 2  |-  ( A 
\  ( C  u.  B ) )  =  ( ( A  \  C )  \  B
)
52, 3, 43eqtr3i 2504 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    u. cun 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483
This theorem is referenced by:  difdifdir  3914  difsnen  7596  cusgrares  24148
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