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Theorem dif32 3605
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 3492 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21difeq2i 3463 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( C  u.  B )
)
3 difun1 3602 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
4 difun1 3602 . 2  |-  ( A 
\  ( C  u.  B ) )  =  ( ( A  \  C )  \  B
)
52, 3, 43eqtr3i 2465 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3318    u. cun 3319
This theorem is referenced by:  difdifdir  3716  difsnen  7191  cusgrares  21482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328
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