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Theorem eqsstrid 3201
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrid.1  |-  A  =  B
eqsstrid.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
eqsstrid  |-  ( ph  ->  A  C_  C )

Proof of Theorem eqsstrid
StepHypRef Expression
1 eqsstrid.2 . 2  |-  ( ph  ->  B  C_  C )
2 eqsstrid.1 . . 3  |-  A  =  B
32sseq1i 3181 . 2  |-  ( A 
C_  C  <->  B  C_  C
)
41, 3sylibr 134 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  eqsstrrid  3202  inss  3365  difsnss  3737  tpssi  3757  peano5  4594  xpsspw  4735  iotanul  5189  iotass  5191  fun  5384  fun11iun  5478  fvss  5525  fmpt  5662  fliftrel  5787  opabbrex  5913  1stcof  6158  2ndcof  6159  tfrlemibacc  6321  tfrlemibfn  6323  tfr1onlemssrecs  6334  tfr1onlembacc  6337  tfr1onlembfn  6339  tfrcllemssrecs  6347  tfrcllembacc  6350  tfrcllembfn  6352  caucvgprlemladdrl  7665  peano5nnnn  7879  peano5nni  8908  un0addcl  9195  un0mulcl  9196  strleund  12541  mgmidsssn0  12692  cnptopco  13382  cnconst2  13393  xmetresbl  13600  blsscls2  13653  bj-omtrans  14357
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