Theorem dif32 3400

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 3281	. . 3 |- ( B u. C ) = ( C u. B )
2	1	difeq2i 3252	. 2 |- ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3		difun1 3397	. 2 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4		difun1 3397	. 2 |- ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5	2, 3, 4	3eqtr3i 2206	1 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Syntax hints:   = wceq 1353   \ cdif 3128   u. cun 3129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137

This theorem is referenced by:  difdifdirss  3509