Theorem eqsstrid 3239

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 3219	. 2 |- ( A C_ C <-> B C_ C )
4	1, 3	sylibr 134	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179

This theorem is referenced by:  eqsstrrid  3240  inss  3403  difsnss  3779  tpssi  3800  peano5  4646  xpsspw  4787  iotanul  5247  iotass  5249  fun  5448  fun11iun  5543  fvss  5590  fmpt  5730  fliftrel  5861  ovssunirng  5979  opabbrex  5989  1stcof  6249  2ndcof  6250  tfrlemibacc  6412  tfrlemibfn  6414  tfr1onlemssrecs  6425  tfr1onlembacc  6428  tfr1onlembfn  6430  tfrcllemssrecs  6438  tfrcllembacc  6441  tfrcllembfn  6443  caucvgprlemladdrl  7791  peano5nnnn  8005  peano5nni  9039  un0addcl  9328  un0mulcl  9329  4sqlemafi  12718  4sqlemffi  12719  4sqleminfi  12720  4sqlem11  12724  4sqlem19  12732  strleund  12935  mgmidsssn0  13216  lsptpcl  14156  cnptopco  14694  cnconst2  14705  xmetresbl  14912  blsscls2  14965  perfectlem2  15472  bj-omtrans  15892