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Theorem ssbrd 3834
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ssbrd.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
ssbrd  |-  ( ph  ->  ( C A D  ->  C B D ) )

Proof of Theorem ssbrd
StepHypRef Expression
1 ssbrd.1 . . 3  |-  ( ph  ->  A  C_  B )
21sseld 2999 . 2  |-  ( ph  ->  ( <. C ,  D >.  e.  A  ->  <. C ,  D >.  e.  B ) )
3 df-br 3794 . 2  |-  ( C A D  <->  <. C ,  D >.  e.  A )
4 df-br 3794 . 2  |-  ( C B D  <->  <. C ,  D >.  e.  B )
52, 3, 43imtr4g 203 1  |-  ( ph  ->  ( C A D  ->  C B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434    C_ wss 2974   <.cop 3409   class class class wbr 3793
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-br 3794
This theorem is referenced by:  ssbri  3835  sess1  4100  brrelex12  4407  coss1  4519  coss2  4520  eqbrrdva  4533  ersym  6184  ertr  6187
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