Theorem sseqtri 3213

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 3206	. 2 |- ( A C_ B <-> A C_ C )
4	1, 3	mpbi 145	1 |- A C_ C

Syntax hints:   = wceq 1364   C_ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  sseqtrri  3214  eqimssi  3235  abssi  3254  ssun2  3323  inssddif  3400  difdifdirss  3531  ifidss  3572  pwundifss  4316  unixpss  4772  0ima  5025  sbthlem7  7022  ssnnctlemct  12603  toponsspwpwg  14190  eltg4i  14223  ntrss2  14289  isopn3  14293  tgioo  14714  dvfvalap  14835  dvcnp2cntop  14848