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Theorem dif32 3371
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 3252 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21difeq2i 3223 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( C  u.  B )
)
3 difun1 3368 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
4 difun1 3368 . 2  |-  ( A 
\  ( C  u.  B ) )  =  ( ( A  \  C )  \  B
)
52, 3, 43eqtr3i 2186 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1335    \ cdif 3099    u. cun 3100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108
This theorem is referenced by:  difdifdirss  3479
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