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Theorem syl5eqss 2989
 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 2969 . 2 (𝐴𝐶𝐵𝐶)
41, 3sylibr 137 1 (𝜑𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931 This theorem is referenced by:  syl5eqssr  2990  inss  3166  difsnss  3510  tpssi  3530  peano5  4321  xpsspw  4450  iotanul  4882  iotass  4884  fun  5063  fun11iun  5147  fvss  5189  fmpt  5319  fliftrel  5432  opabbrex  5549  1stcof  5790  2ndcof  5791  tfrlemibacc  5940  tfrlemibfn  5942  caucvgprlemladdrl  6776  peano5nnnn  6966  peano5nni  7917  un0addcl  8215  un0mulcl  8216  peano5setOLD  10065  bj-omtrans  10081
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