Theorem sseqtri 2971

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 2964	. 2 ⊢ (A ⊆ B ↔ A ⊆ 𝐶)
4	1, 3	mpbi 133	1 ⊢ A ⊆ 𝐶

Syntax hints:   = wceq 1242   ⊆ wss 2911

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  sseqtr4i  2972  eqimssi  2993  abssi  3009  ssun2  3101  inssddif  3172  difdifdirss  3301  pwundifss  4013  unixpss  4394  0ima  4628