Theorem sseqtri 3181

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 3174	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	mpbi 144	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1348   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  sseqtrri  3182  eqimssi  3203  abssi  3222  ssun2  3291  inssddif  3368  difdifdirss  3499  ifidss  3541  pwundifss  4270  unixpss  4724  0ima  4971  sbthlem7  6940  ssnnctlemct  12401  toponsspwpwg  12814  eltg4i  12849  ntrss2  12915  isopn3  12919  tgioo  13340  dvfvalap  13444  dvcnp2cntop  13457