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Theorem ssdifd 3483
Description: If  A is contained in  B, then  ( A  \  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3482. (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
StepHypRef Expression
1 ssdifd.1 . 2  |-  ( ph  ->  A  C_  B )
2 ssdif 3482 . 2  |-  ( A 
C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
31, 2syl 16 1  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3317    C_ wss 3320
This theorem is referenced by:  ssdif2d  3486  domunsncan  7208  fin1a2lem13  8292  seqcoll2  11713  rpnnen2lem11  12824  mrieqv2d  13864  mrissmrid  13866  mreexexlem4d  13872  acsfiindd  14603  lsppratlem3  16221  lsppratlem4  16222  lpss3  17208  lpcls  17428  fin1aufil  17964  uniioombllem3  19477  i1fmul  19588  itg1addlem4  19591  itg1climres  19606  limciun  19781  ig1peu  20094  ig1pdvds  20099  nbgrassovt  21447  sitgclg  24656  f1lindf  27269  usgreghash2spotv  28455  dochfln0  32275  lcfl6  32298  lcfrlem16  32356  hdmaprnlem4N  32654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334
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