Theorem sseqtri 3175

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 3168	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 144	1 |- A C_ C

Syntax hints:   = wceq 1343   C_ wss 3115

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3121  df-ss 3128

This theorem is referenced by:  sseqtrri  3176  eqimssi  3197  abssi  3216  ssun2  3285  inssddif  3362  difdifdirss  3492  ifidss  3534  pwundifss  4262  unixpss  4716  0ima  4963  sbthlem7  6924  ssnnctlemct  12375  toponsspwpwg  12620  eltg4i  12655  ntrss2  12721  isopn3  12725  tgioo  13146  dvfvalap  13250  dvcnp2cntop  13263