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Theorem dif32 3713
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 3587 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21difeq2i 3558 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( C  u.  B )
)
3 difun1 3710 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
4 difun1 3710 . 2  |-  ( A 
\  ( C  u.  B ) )  =  ( ( A  \  C )  \  B
)
52, 3, 43eqtr3i 2439 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    \ cdif 3411    u. cun 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421
This theorem is referenced by:  difdifdir  3859  difsnen  7637  cusgrares  24889
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