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 )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

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 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 theorem 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 referenced by:  sseqtrrdi  3297  onintonm  4659  relrelss  5309  iotanul  5348  foimacnv  5652  pw1m  7573  cauappcvgprlemladdru  8013  nninfdcex  10650  zsupssdc  10651  hashfibclem  11260  zsumdc  12129  fsum3cvg3  12141  zproddc  12324  imasaddfnlemg  13612  sraring  14758  distop  15109  cnptoprest  15263  upgr1edc  16276  uspgr1edc  16395  pw1ndom3lem  16933  pwle2  16942  pw1nct  16947