Theorem sseqtri 3274

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 3267	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1398   ⊆ wss 3213

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 3219  df-ss 3226

This theorem is referenced by:  sseqtrri  3275  eqimssi  3296  abssi  3315  ssun2  3385  inssddif  3464  difdifdirss  3596  ifidss  3640  pwundifss  4408  unixpss  4865  0ima  5124  sbthlem7  7235  0bits  12653  ssnnctlemct  13218  prdsvallem  13506  toponsspwpwg  14936  eltg4i  14969  ntrss2  15035  isopn3  15039  tgioo  15468  dvfvalap  15595  dvcnp2cntop  15613