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Theorem ssbrd 4064
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 3179 . 2  |-  ( ph  ->  ( <. C ,  D >.  e.  A  ->  <. C ,  D >.  e.  B ) )
3 df-br 4024 . 2  |-  ( C A D  <->  <. C ,  D >.  e.  A )
4 df-br 4024 . 2  |-  ( C B D  <->  <. C ,  D >.  e.  B )
52, 3, 43imtr4g 261 1  |-  ( ph  ->  ( C A D  ->  C B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023
This theorem is referenced by:  ssbri  4065  sess1  4361  brrelex12  4726  coss1  4839  coss2  4840  ersym  6672  ertr  6675  fpwwe2lem6  8257  fpwwe2lem7  8258  fpwwe2lem9  8260  fpwwe2lem12  8263  fpwwe2lem13  8264  fpwwe2  8265  fthres2  13806  invfuc  13848  pospo  14107  dirref  14357  efgcpbl  15065  frgpuplem  15081  subrguss  15560  znleval  16508  fundmpss  24122
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-br 4024
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