Theorem sseqtri 3231

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 3224	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 145	1 |- A C_ C

Syntax hints:   = wceq 1373   C_ wss 3170

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3176  df-ss 3183

This theorem is referenced by:  sseqtrri  3232  eqimssi  3253  abssi  3272  ssun2  3341  inssddif  3418  difdifdirss  3549  ifidss  3591  pwundifss  4340  unixpss  4796  0ima  5051  sbthlem7  7080  0bits  12345  ssnnctlemct  12892  prdsvallem  13179  toponsspwpwg  14569  eltg4i  14602  ntrss2  14668  isopn3  14672  tgioo  15101  dvfvalap  15228  dvcnp2cntop  15246