Theorem eqsstrid 3270

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 3250	. 2 |- ( A C_ C <-> B C_ C )
4	1, 3	sylibr 134	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  eqsstrrid  3271  inss  3434  difsnss  3814  tpssi  3837  peano5  4690  xpsspw  4831  iotanul  5294  iotass  5296  fun  5497  fun11iun  5593  fvss  5641  fmpt  5785  fliftrel  5916  ovssunirng  6036  opabbrex  6048  1stcof  6309  2ndcof  6310  tfrlemibacc  6472  tfrlemibfn  6474  tfr1onlemssrecs  6485  tfr1onlembacc  6488  tfr1onlembfn  6490  tfrcllemssrecs  6498  tfrcllembacc  6501  tfrcllembfn  6503  caucvgprlemladdrl  7865  peano5nnnn  8079  peano5nni  9113  un0addcl  9402  un0mulcl  9403  4sqlemafi  12918  4sqlemffi  12919  4sqleminfi  12920  4sqlem11  12924  4sqlem19  12932  strleund  13136  mgmidsssn0  13417  lsptpcl  14358  cnptopco  14896  cnconst2  14907  xmetresbl  15114  blsscls2  15167  perfectlem2  15674  setsvtx  15852  bj-omtrans  16319