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Theorem syl5eqss 3059
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 3039 . 2 (𝐴𝐶𝐵𝐶)
41, 3sylibr 132 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-in 2994  df-ss 3001
This theorem is referenced by:  syl5eqssr  3060  inss  3218  difsnss  3566  tpssi  3586  peano5  4385  xpsspw  4517  iotanul  4958  iotass  4960  fun  5141  fun11iun  5231  fvss  5276  fmpt  5406  fliftrel  5526  opabbrex  5644  1stcof  5885  2ndcof  5886  tfrlemibacc  6039  tfrlemibfn  6041  tfr1onlemssrecs  6052  tfr1onlembacc  6055  tfr1onlembfn  6057  tfrcllemssrecs  6065  tfrcllembacc  6068  tfrcllembfn  6070  caucvgprlemladdrl  7174  peano5nnnn  7364  peano5nni  8353  un0addcl  8632  un0mulcl  8633  bj-omtrans  11281
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