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  4664  relrelss  5314  iotanul  5353  foimacnv  5657  pw1m  7584  cauappcvgprlemladdru  8024  nninfdcex  10683  zsupssdc  10684  hashfibclem  11298  zsumdc  12170  fsum3cvg3  12182  zproddc  12365  imasaddfnlemg  13688  sraring  14870  distop  15277  cnptoprest  15431  upgr1edc  16528  uspgr1edc  16647  pw1ndom3lem  17185  pwle2  17194  pw1nct  17199