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Theorem ssdif2d 3315
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 3313 . 2  |-  ( ph  ->  ( A  \  D
)  C_  ( A  \  C ) )
3 ssdifd.1 . . 3  |-  ( ph  ->  A  C_  B )
43ssdifd 3312 . 2  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
52, 4sstrd 3189 1  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3149    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166
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