Theorem ssdifd 3633

Description: If A is contained in B, then ( A \ C ) is contained in ( B \ C ). Deduction form of ssdif 3632. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	|- ( ph -> A C_ B )

Assertion

Ref	Expression
ssdifd	|- ( ph -> ( A \ C ) C_ ( B \ C ) )

Proof of Theorem ssdifd

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 |- ( ph -> A C_ B )
2		ssdif 3632	. 2 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )
3	1, 2	syl 16	1 |- ( ph -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3466   C_ wss 3469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483

This theorem is referenced by:  ssdif2d  3636  domunsncan  7607  fin1a2lem13  8781  seqcoll2  12466  rpnnen2lem11  13808  mrieqv2d  14883  mrissmrid  14885  mreexexlem4d  14891  acsfiindd  15653  lsppratlem3  17571  lsppratlem4  17572  f1lindf  18617  lpss3  19404  lpcls  19624  fin1aufil  20161  rrxmval  21560  rrxmetlem  21562  uniioombllem3  21722  i1fmul  21831  itg1addlem4  21834  itg1climres  21849  limciun  22026  ig1peu  22300  ig1pdvds  22305  nbgrassovt  24097  usgreghash2spotv  24729  sitgclg  27910  dochfln0  36149  lcfl6  36172  lcfrlem16  36230  hdmaprnlem4N  36528