Theorem sseqtri 3282

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 3275	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 145	1 |- A C_ C

This proof depends on syntax axioms:   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  sseqtrri  3283  eqimssi  3304  abssi  3323  ssun2  3393  inssddif  3472  difdifdirss  3612  ifidss  3656  pwundifss  4430  unixpss  4888  0ima  5147  rinvf1o  6035  sbthlem7  7280  0bits  12726  ssnnctlemct  13337  prdsvallem  13621  toponsspwpwg  15123  eltg4i  15156  ntrss2  15222  isopn3  15226  tgioo  15655  dvfvalap  15782  dvcnp2cntop  15800