Theorem sseqtrdi 3296

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 3275	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	sylib 122	1 |- ( ph -> A C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  sseqtrrdi  3297  onintonm  4666  relrelss  5317  iotanul  5356  foimacnv  5660  pw1m  7589  cauappcvgprlemladdru  8029  nninfdcex  10688  zsupssdc  10689  hashfibclem  11303  zsumdc  12175  fsum3cvg3  12187  zproddc  12370  imasaddfnlemg  13693  sraring  14894  distop  15328  cnptoprest  15482  upgr1edc  16584  uspgr1edc  16703  pw1ndom3lem  17241  pwle2  17250  pw1nct  17255