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Theorem syl5eqss 2962
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqss.1 A = B
syl5eqss.2 (φB𝐶)
Assertion
Ref Expression
syl5eqss (φA𝐶)

Proof of Theorem syl5eqss
StepHypRef Expression
1 syl5eqss.2 . 2 (φB𝐶)
2 syl5eqss.1 . . 3 A = B
32sseq1i 2942 . 2 (A𝐶B𝐶)
41, 3sylibr 137 1 (φA𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226  wss 2890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904
This theorem is referenced by:  syl5eqssr  2963  inss  3139  difsnss  3480  tpssi  3500  peano5  4244  xpsspw  4373  iotanul  4805  iotass  4807  fun  4984  fun11iun  5068  fvss  5110  fmpt  5240  fliftrel  5353  opabbrex  5468  1stcof  5709  2ndcof  5710  tfrlemibacc  5857  tfrlemibfn  5859  peano5set  7301  bj-omtrans  7317
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