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Theorem syl5eqss 3044
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 3024 . 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 1285    C_ wss 2974
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  syl5eqssr  3045  inss  3202  difsnss  3539  tpssi  3559  peano5  4347  xpsspw  4478  iotanul  4912  iotass  4914  fun  5094  fun11iun  5178  fvss  5220  fmpt  5351  fliftrel  5463  opabbrex  5580  1stcof  5821  2ndcof  5822  tfrlemibacc  5975  tfrlemibfn  5977  tfr1onlemssrecs  5988  tfr1onlembacc  5991  tfr1onlembfn  5993  tfrcllemssrecs  6001  tfrcllembacc  6004  tfrcllembfn  6006  caucvgprlemladdrl  6930  peano5nnnn  7120  peano5nni  8109  un0addcl  8388  un0mulcl  8389  bj-omtrans  10909
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