Theorem eqsstrid 3225

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 3205	. 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 3153

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 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  eqsstrrid  3226  inss  3389  difsnss  3764  tpssi  3785  peano5  4630  xpsspw  4771  iotanul  5230  iotass  5232  fun  5426  fun11iun  5521  fvss  5568  fmpt  5708  fliftrel  5835  opabbrex  5962  1stcof  6216  2ndcof  6217  tfrlemibacc  6379  tfrlemibfn  6381  tfr1onlemssrecs  6392  tfr1onlembacc  6395  tfr1onlembfn  6397  tfrcllemssrecs  6405  tfrcllembacc  6408  tfrcllembfn  6410  caucvgprlemladdrl  7738  peano5nnnn  7952  peano5nni  8985  un0addcl  9273  un0mulcl  9274  4sqlemafi  12533  4sqlemffi  12534  4sqleminfi  12535  4sqlem11  12539  4sqlem19  12547  strleund  12721  mgmidsssn0  12967  lsptpcl  13890  cnptopco  14390  cnconst2  14401  xmetresbl  14608  blsscls2  14661  bj-omtrans  15448