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Theorem ssbrd 4080
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 3192 . 2  |-  ( ph  ->  ( <. C ,  D >.  e.  A  ->  <. C ,  D >.  e.  B ) )
3 df-br 4040 . 2  |-  ( C A D  <->  <. C ,  D >.  e.  A )
4 df-br 4040 . 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 1696    C_ wss 3165   <.cop 3656   class class class wbr 4039
This theorem is referenced by:  ssbri  4081  sess1  4377  brrelex12  4742  coss1  4855  coss2  4856  ersym  6688  ertr  6691  fpwwe2lem6  8273  fpwwe2lem7  8274  fpwwe2lem9  8276  fpwwe2lem12  8279  fpwwe2lem13  8280  fpwwe2  8281  fthres2  13822  invfuc  13864  pospo  14123  dirref  14373  efgcpbl  15081  frgpuplem  15097  subrguss  15576  znleval  16524  fundmpss  24193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-br 4040
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