Theorem sseqtrdi 3276

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 3255	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 122	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  sseqtrrdi  3277  onintonm  4621  relrelss  5270  iotanul  5309  foimacnv  5610  pw1m  7485  cauappcvgprlemladdru  7919  nninfdcex  10543  zsupssdc  10544  zsumdc  12008  fsum3cvg3  12020  zproddc  12203  imasaddfnlemg  13460  sraring  14528  distop  14879  cnptoprest  15033  upgr1edc  16045  uspgr1edc  16164  pw1ndom3lem  16692  pwle2  16703  pw1nct  16708