Theorem sseqtri 3258

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 3251	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 145	1 |- A C_ C

Syntax hints:   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  sseqtrri  3259  eqimssi  3280  abssi  3299  ssun2  3368  inssddif  3445  difdifdirss  3576  ifidss  3618  pwundifss  4376  unixpss  4832  0ima  5088  sbthlem7  7141  0bits  12485  ssnnctlemct  13032  prdsvallem  13320  toponsspwpwg  14711  eltg4i  14744  ntrss2  14810  isopn3  14814  tgioo  15243  dvfvalap  15370  dvcnp2cntop  15388