Theorem dif32 3200

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 3087	. . 3 |- ( B u. C ) = ( C u. B )
2	1	difeq2i 3059	. 2 |- ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3		difun1 3197	. 2 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4		difun1 3197	. 2 |- ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5	2, 3, 4	3eqtr3i 2068	1 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Syntax hints:   = wceq 1243   \ cdif 2914   u. cun 2915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924

This theorem is referenced by:  difdifdirss  3307