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Theorem dif32 3307
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 3188 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
21difeq2i 3159 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( A  \  ( C  u.  B )
)
3 difun1 3304 . 2  |-  ( A 
\  ( B  u.  C ) )  =  ( ( A  \  B )  \  C
)
4 difun1 3304 . 2  |-  ( A 
\  ( C  u.  B ) )  =  ( ( A  \  C )  \  B
)
52, 3, 43eqtr3i 2144 1  |-  ( ( A  \  B ) 
\  C )  =  ( ( A  \  C )  \  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1314    \ cdif 3036    u. cun 3037
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045
This theorem is referenced by:  difdifdirss  3415
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