Theorem sseqtri 2977

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 2970	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	mpbi 133	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   = wceq 1243   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  sseqtr4i  2978  eqimssi  2999  abssi  3015  ssun2  3107  inssddif  3178  difdifdirss  3307  pwundifss  4022  unixpss  4451  0ima  4685