Theorem sseqtri 3162

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Hypotheses

Ref	Expression
sseqtr.1	|- A C_ B
sseqtr.2	|- B = C

Assertion

Ref	Expression
sseqtri	|- A C_ C

Proof of Theorem sseqtri

Step	Hyp	Ref	Expression
1		sseqtr.1	. 2 |- A C_ B
2		sseqtr.2	. . 3 |- B = C
3	2	sseq2i 3155	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 144	1 |- A C_ C

Syntax hints:   = wceq 1335   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115

This theorem is referenced by:  sseqtrri  3163  eqimssi  3184  abssi  3203  ssun2  3271  inssddif  3348  difdifdirss  3478  ifidss  3520  pwundifss  4246  unixpss  4700  0ima  4947  sbthlem7  6908  ssnnctlemct  12217  toponsspwpwg  12462  eltg4i  12497  ntrss2  12563  isopn3  12567  tgioo  12988  dvfvalap  13092  dvcnp2cntop  13105