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Theorem ssbrd 3934
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 3060 . 2  |-  ( ph  ->  ( <. C ,  D >.  e.  A  ->  <. C ,  D >.  e.  B ) )
3 df-br 3894 . 2  |-  ( C A D  <->  <. C ,  D >.  e.  A )
4 df-br 3894 . 2  |-  ( C B D  <->  <. C ,  D >.  e.  B )
52, 3, 43imtr4g 204 1  |-  ( ph  ->  ( C A D  ->  C B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1461    C_ wss 3035   <.cop 3494   class class class wbr 3893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-in 3041  df-ss 3048  df-br 3894
This theorem is referenced by:  ssbri  3935  sess1  4217  brrelex12  4535  coss1  4652  coss2  4653  eqbrrdva  4667  ersym  6393  ertr  6396
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