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Theorem ssdif2d 3286
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 3284 . 2  |-  ( ph  ->  ( A  \  D
)  C_  ( A  \  C ) )
3 ssdifd.1 . . 3  |-  ( ph  ->  A  C_  B )
43ssdifd 3283 . 2  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
52, 4sstrd 3177 1  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3138    C_ wss 3141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154
This theorem is referenced by: (None)
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