Theorem sseqtrdi 3273

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 3252	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 122	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  sseqtrrdi  3274  onintonm  4613  relrelss  5261  iotanul  5300  foimacnv  5598  pw1m  7432  cauappcvgprlemladdru  7866  nninfdcex  10487  zsupssdc  10488  zsumdc  11935  fsum3cvg3  11947  zproddc  12130  imasaddfnlemg  13387  sraring  14453  distop  14799  cnptoprest  14953  upgr1edc  15962  uspgr1edc  16079  pw1ndom3lem  16524  pwle2  16535  pw1nct  16540