Theorem dif32 3436

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 3317	. . 3 |- ( B u. C ) = ( C u. B )
2	1	difeq2i 3288	. 2 |- ( A \ ( B u. C ) ) = ( A \ ( C u. B ) )
3		difun1 3433	. 2 |- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
4		difun1 3433	. 2 |- ( A \ ( C u. B ) ) = ( ( A \ C ) \ B )
5	2, 3, 4	3eqtr3i 2234	1 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )

Syntax hints:   = wceq 1373   \ cdif 3163   u. cun 3164

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172

This theorem is referenced by:  difdifdirss  3545