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Theorem syl5eqss 2986
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
syl5eqss.1  |-  A  =  B
syl5eqss.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
syl5eqss  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl5eqss
StepHypRef Expression
1 syl5eqss.2 . 2  |-  ( ph  ->  B  C_  C )
2 syl5eqss.1 . . 3  |-  A  =  B
32sseq1i 2966 . 2  |-  ( A 
C_  C  <->  B  C_  C
)
41, 3sylibr 137 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    C_ wss 2914
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 2921  df-ss 2928
This theorem is referenced by:  syl5eqssr  2987  inss  3163  difsnss  3507  tpssi  3527  peano5  4299  xpsspw  4428  iotanul  4860  iotass  4862  fun  5041  fun11iun  5125  fvss  5167  fmpt  5297  fliftrel  5410  opabbrex  5527  1stcof  5768  2ndcof  5769  tfrlemibacc  5918  tfrlemibfn  5920  caucvgprlemladdrl  6748  peano5nnnn  6938  peano5nni  7884  un0addcl  8178  un0mulcl  8179  peano5setOLD  9929  bj-omtrans  9945
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