Theorem syl5eqss 3043

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl5eqss.1	|- A = B
syl5eqss.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
syl5eqss	|- ( ph -> A C_ C )

Proof of Theorem syl5eqss

Step	Hyp	Ref	Expression
1		syl5eqss.2	. 2 |- ( ph -> B C_ C )
2		syl5eqss.1	. . 3 |- A = B
3	2	sseq1i 3023	. 2 |- ( A C_ C <-> B C_ C )
4	1, 3	sylibr 132	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1284   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  syl5eqssr  3044  inss  3195  difsnss  3531  tpssi  3551  peano5  4339  xpsspw  4468  iotanul  4902  iotass  4904  fun  5083  fun11iun  5167  fvss  5209  fmpt  5340  fliftrel  5452  opabbrex  5569  1stcof  5810  2ndcof  5811  tfrlemibacc  5963  tfrlemibfn  5965  caucvgprlemladdrl  6868  peano5nnnn  7058  peano5nni  8042  un0addcl  8321  un0mulcl  8322  bj-omtrans  10751