Theorem difcom 4053

Description: Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)

Assertion

Ref	Expression
difcom	|- ( ( A \ B ) C_ C <-> ( A \ C ) C_ B )

Proof of Theorem difcom

Step	Hyp	Ref	Expression
1		uncom 3757	. . 3 |- ( B u. C ) = ( C u. B )
2	1	sseq2i 3630	. 2 |- ( A C_ ( B u. C ) <-> A C_ ( C u. B ) )
3		ssundif 4052	. 2 |- ( A C_ ( B u. C ) <-> ( A \ B ) C_ C )
4		ssundif 4052	. 2 |- ( A C_ ( C u. B ) <-> ( A \ C ) C_ B )
5	2, 3, 4	3bitr3i 290	1 |- ( ( A \ B ) C_ C <-> ( A \ C ) C_ B )

Syntax hints:   <-> wb 196   \ cdif 3571   u. cun 3572   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by:  pssdifcom1  4054  pssdifcom2  4055  isreg2  21181  restmetu  22375  conss1  38648  icccncfext  40100