Theorem eqsstrid 3273

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 3253	. 2 |- ( A C_ C <-> B C_ C )
4	1, 3	sylibr 134	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1397   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  eqsstrrid  3274  inss  3437  difsnss  3819  tpssi  3842  peano5  4696  opabssxpd  4762  xpsspw  4838  iotanul  5302  iotass  5304  fun  5508  fun11iun  5604  fvss  5653  fmpt  5797  fliftrel  5933  ovssunirng  6053  opabbrex  6065  1stcof  6326  2ndcof  6327  tfrlemibacc  6492  tfrlemibfn  6494  tfr1onlemssrecs  6505  tfr1onlembacc  6508  tfr1onlembfn  6510  tfrcllemssrecs  6518  tfrcllembacc  6521  tfrcllembfn  6523  caucvgprlemladdrl  7898  peano5nnnn  8112  peano5nni  9146  un0addcl  9435  un0mulcl  9436  4sqlemafi  12986  4sqlemffi  12987  4sqleminfi  12988  4sqlem11  12992  4sqlem19  13000  strleund  13204  mgmidsssn0  13485  lsptpcl  14427  cnptopco  14965  cnconst2  14976  xmetresbl  15183  blsscls2  15236  perfectlem2  15743  setsvtx  15921  1hegrvtxdg1rfi  16180  bj-omtrans  16602