Theorem sseqtri 3276

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Hypotheses

Ref	Expression
sseqtr.1	⊢ 𝐴 ⊆ 𝐵
sseqtr.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
sseqtri	⊢ 𝐴 ⊆ 𝐶

Proof of Theorem sseqtri

Step	Hyp	Ref	Expression
1		sseqtr.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		sseqtr.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	sseq2i 3269	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1398   ⊆ wss 3214

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  sseqtrri  3277  eqimssi  3298  abssi  3317  ssun2  3387  inssddif  3466  difdifdirss  3598  ifidss  3642  pwundifss  4411  unixpss  4868  0ima  5127  rinvf1o  6008  sbthlem7  7246  0bits  12673  ssnnctlemct  13284  prdsvallem  13567  toponsspwpwg  15016  eltg4i  15049  ntrss2  15115  isopn3  15119  tgioo  15548  dvfvalap  15675  dvcnp2cntop  15693