Theorem ssdifd 3746

Description: If A is contained in B, then ( A \ C ) is contained in ( B \ C ). Deduction form of ssdif 3745. (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 3745	. 2 |- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )
3	1, 2	syl 17	1 |- ( ph -> ( A \ C ) C_ ( B \ C ) )

Syntax hints:   -> wi 4   \ cdif 3571   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-in 3581  df-ss 3588

This theorem is referenced by:  ssdif2d  3749  domunsncan  8060  fin1a2lem13  9234  seqcoll2  13249  rpnnen2lem11  14953  coprmprod  15375  mrieqv2d  16299  mrissmrid  16301  mreexexlem4d  16307  acsfiindd  17177  lsppratlem3  19149  lsppratlem4  19150  f1lindf  20161  lpss3  20948  lpcls  21168  fin1aufil  21736  rrxmval  23188  rrxmetlem  23190  uniioombllem3  23353  i1fmul  23463  itg1addlem4  23466  itg1climres  23481  limciun  23658  ig1peu  23931  ig1pdvds  23936  fusgreghash2wspv  27199  indsumin  30084  sitgclg  30404  mthmpps  31479  poimirlem11  33420  poimirlem12  33421  poimirlem15  33424  dochfln0  36766  lcfl6  36789  lcfrlem16  36847  hdmaprnlem4N  37145  caragendifcl  40728