Theorem dif32 3385

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 3266	. . 3 |- ( B u. C ) = ( C u. B )
2	1	difeq2i 3237	. 2 |- ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3		difun1 3382	. 2 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4		difun1 3382	. 2 |- ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5	2, 3, 4	3eqtr3i 2194	1 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Syntax hints:   = wceq 1343   \ cdif 3113   u. cun 3114

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122

This theorem is referenced by:  difdifdirss  3493