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Theorem syl5eqssr 3236
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqssr.1  |-  B  =  A
syl5eqssr.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
syl5eqssr  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl5eqssr
StepHypRef Expression
1 syl5eqssr.1 . . 3  |-  B  =  A
21eqcomi 2300 . 2  |-  A  =  B
3 syl5eqssr.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3syl5eqss 3235 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165
This theorem is referenced by:  relcnvtr  5208  fimacnvdisj  5435  dffv2  5608  fimacnv  5673  f1ompt  5698  fnwelem  6246  tfrlem15  6424  omxpenlem  6979  hartogslem1  7273  infxpidm2  7660  alephgeom  7725  infenaleph  7734  cfflb  7901  pwfseqlem5  8301  imasvscafn  13455  mrieqvlemd  13547  cnvps  14337  dirdm  14372  tsrdir  14376  frmdss2  14501  iinopn  16664  xkococnlem  17369  tgpconcomp  17811  mbfconstlem  19000  itg2seq  19113  limcdif  19242  dvres2lem  19276  c1lip3  19362  lhop  19379  plyeq0  19609  dchrghm  20511  chssoc  22091  cvmliftmolem1  23827  cvmlift2lem9a  23849  cvmlift2lem9  23857  cartarlim  26008  pgapspf  26155  cnres2  26586  rngunsnply  27481  proot1hash  27622
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
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