Theorem sseqtri 3228

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 3221	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1373   ⊆ wss 3167

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3173  df-ss 3180

This theorem is referenced by:  sseqtrri  3229  eqimssi  3250  abssi  3269  ssun2  3338  inssddif  3415  difdifdirss  3546  ifidss  3587  pwundifss  4336  unixpss  4792  0ima  5047  sbthlem7  7072  0bits  12314  ssnnctlemct  12861  prdsvallem  13148  toponsspwpwg  14538  eltg4i  14571  ntrss2  14637  isopn3  14641  tgioo  15070  dvfvalap  15197  dvcnp2cntop  15215