Theorem sseqtri 3136

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 3129	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 144	1 |- A C_ C

Syntax hints:   = wceq 1332   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  sseqtrri  3137  eqimssi  3158  abssi  3177  ssun2  3245  inssddif  3322  difdifdirss  3452  pwundifss  4215  unixpss  4660  0ima  4907  sbthlem7  6859  toponsspwpwg  12228  eltg4i  12263  ntrss2  12329  isopn3  12333  tgioo  12754  dvfvalap  12858  dvcnp2cntop  12871