Theorem eqsstrid 3216

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

Step	Hyp	Ref	Expression
1		eqsstrid.2	. 2 |- ( ph -> B C_ C )
2		eqsstrid.1	. . 3 |- A = B
3	2	sseq1i 3196	. 2 |- ( A C_ C <-> B C_ C )
4	1, 3	sylibr 134	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  eqsstrrid  3217  inss  3380  difsnss  3753  tpssi  3774  peano5  4615  xpsspw  4756  iotanul  5211  iotass  5213  fun  5407  fun11iun  5501  fvss  5548  fmpt  5687  fliftrel  5814  opabbrex  5941  1stcof  6189  2ndcof  6190  tfrlemibacc  6352  tfrlemibfn  6354  tfr1onlemssrecs  6365  tfr1onlembacc  6368  tfr1onlembfn  6370  tfrcllemssrecs  6378  tfrcllembacc  6381  tfrcllembfn  6383  caucvgprlemladdrl  7708  peano5nnnn  7922  peano5nni  8953  un0addcl  9240  un0mulcl  9241  4sqlemafi  12430  4sqlemffi  12431  4sqleminfi  12432  4sqlem11  12436  4sqlem19  12444  strleund  12618  mgmidsssn0  12863  lsptpcl  13727  cnptopco  14199  cnconst2  14210  xmetresbl  14417  blsscls2  14470  bj-omtrans  15186