Theorem sseqtri 3259

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 3252	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 145	1 |- A C_ C

Syntax hints:   = wceq 1395   C_ wss 3198

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 3204  df-ss 3211

This theorem is referenced by:  sseqtrri  3260  eqimssi  3281  abssi  3300  ssun2  3369  inssddif  3446  difdifdirss  3577  ifidss  3619  pwundifss  4380  unixpss  4837  0ima  5094  sbthlem7  7153  0bits  12510  ssnnctlemct  13057  prdsvallem  13345  toponsspwpwg  14736  eltg4i  14769  ntrss2  14835  isopn3  14839  tgioo  15268  dvfvalap  15395  dvcnp2cntop  15413