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Theorem ssdif2d 3488
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 3486 . 2  |-  ( ph  ->  ( A  \  D
)  C_  ( A  \  C ) )
3 ssdifd.1 . . 3  |-  ( ph  ->  A  C_  B )
43ssdifd 3485 . 2  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
52, 4sstrd 3360 1  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3319    C_ wss 3322
This theorem is referenced by:  mblfinlem3  26247  mblfinlem4  26248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336
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