Theorem dif32 3227

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

Step	Hyp	Ref	Expression
1		uncom 3116	. . 3 |- ( B u. C ) = ( C u. B )
2	1	difeq2i 3087	. 2 |- ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3		difun1 3224	. 2 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4		difun1 3224	. 2 |- ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5	2, 3, 4	3eqtr3i 2109	1 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Syntax hints:   = wceq 1284   \ cdif 2970   u. cun 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979

This theorem is referenced by:  difdifdirss  3327