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Theorem syl6eqssr 3555
Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
syl6eqssr.1  |-  ( ph  ->  B  =  A )
syl6eqssr.2  |-  B  C_  C
Assertion
Ref Expression
syl6eqssr  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl6eqssr
StepHypRef Expression
1 syl6eqssr.1 . . 3  |-  ( ph  ->  B  =  A )
21eqcomd 2475 . 2  |-  ( ph  ->  A  =  B )
3 syl6eqssr.2 . 2  |-  B  C_  C
42, 3syl6eqss 3554 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490
This theorem is referenced by:  ffvresb  6052  tposss  6956  sbthlem5  7631  rankxpl  8293  winafp  9075  wunex2  9116  iooval2  11562  telfsumo  13579  structcnvcnv  14501  ressbasss  14547  tsrdir  15725  idrespermg  16241  symgsssg  16298  opsrtoslem2  17948  dsmmsubg  18569  cnclsi  19567  txss12  19869  txbasval  19870  kqsat  19995  kqcldsat  19997  fmss  20210  cfilucfilOLD  20835  cfilucfil  20836  tngtopn  20927  dvaddf  22108  dvmulf  22109  dvcof  22114  dvmptres3  22122  dvmptres2  22128  dvmptcj  22134  dvcnvlem  22140  dvcnv  22141  dvcnvrelem1  22181  dvcnvrelem2  22182  plyrem  22463  ulmss  22554  ulmdvlem1  22557  ulmdvlem3  22559  ulmdv  22560  isppw  23144  dchrelbas2  23268  chsupsn  26035  pjss1coi  26786  off2  27182  resspos  27337  resstos  27338  submomnd  27390  suborng  27496  omsmon  27935  signstfvn  28194  hbtlem6  30710
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