Theorem sseqtrdi 3189

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

Hypotheses

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

Assertion

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

Proof of Theorem sseqtrdi

Step	Hyp	Ref	Expression
1		sseqtrdi.1	. 2 |- ( ph -> A C_ B )
2		sseqtrdi.2	. . 3 |- B = C
3	2	sseq2i 3168	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 121	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3121  df-ss 3128

This theorem is referenced by:  sseqtrrdi  3190  onintonm  4493  relrelss  5129  iotanul  5167  foimacnv  5449  cauappcvgprlemladdru  7593  zsumdc  11321  fsum3cvg3  11333  zproddc  11516  nninfdcex  11882  zsupssdc  11883  distop  12685  cnptoprest  12839  pwle2  13838  pw1nct  13843