Theorem sseqtrdi 3176

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 3155	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 121	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1335   C_ wss 3102

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115

This theorem is referenced by:  sseqtrrdi  3177  onintonm  4477  relrelss  5113  iotanul  5151  foimacnv  5433  cauappcvgprlemladdru  7577  zsumdc  11285  fsum3cvg3  11297  zproddc  11480  nninfdcex  11843  distop  12527  cnptoprest  12681  pwle2  13612  pw1nct  13617