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Theorem ssdif2d 3121
Description: If  A is contained in  B and  C is contained in  D, then  ( A  \  D ) is contained in  ( B  \  C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssdifd.1  |-  ( ph  ->  A  C_  B )
ssdif2d.2  |-  ( ph  ->  C  C_  D )
Assertion
Ref Expression
ssdif2d  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )

Proof of Theorem ssdif2d
StepHypRef Expression
1 ssdif2d.2 . . 3  |-  ( ph  ->  C  C_  D )
21sscond 3119 . 2  |-  ( ph  ->  ( A  \  D
)  C_  ( A  \  C ) )
3 ssdifd.1 . . 3  |-  ( ph  ->  A  C_  B )
43ssdifd 3118 . 2  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
52, 4sstrd 3018 1  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 2979    C_ wss 2982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995
This theorem is referenced by: (None)
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