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Theorem syl5eqss 3068
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 3048 . 2  |-  ( A 
C_  C  <->  B  C_  C
)
41, 3sylibr 132 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2997
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010
This theorem is referenced by:  syl5eqssr  3069  inss  3227  difsnss  3578  tpssi  3598  peano5  4403  xpsspw  4538  iotanul  4982  iotass  4984  fun  5168  fun11iun  5258  fvss  5303  fmpt  5433  fliftrel  5553  opabbrex  5675  1stcof  5916  2ndcof  5917  tfrlemibacc  6073  tfrlemibfn  6075  tfr1onlemssrecs  6086  tfr1onlembacc  6089  tfr1onlembfn  6091  tfrcllemssrecs  6099  tfrcllembacc  6102  tfrcllembfn  6104  caucvgprlemladdrl  7216  peano5nnnn  7406  peano5nni  8397  un0addcl  8676  un0mulcl  8677  bj-omtrans  11508
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