Theorem sseqtrdi 3290

Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrdi.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
sseqtrdi.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
sseqtrdi	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sseqtrdi

Step	Hyp	Ref	Expression
1		sseqtrdi.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseqtrdi.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	sseq2i 3269	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	sylib 122	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1398   ⊆ wss 3214

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 3220  df-ss 3227

This theorem is referenced by:  sseqtrrdi  3291  onintonm  4644  relrelss  5294  iotanul  5333  foimacnv  5637  pw1m  7547  cauappcvgprlemladdru  7987  nninfdcex  10624  zsupssdc  10625  hashfibclem  11234  zsumdc  12098  fsum3cvg3  12110  zproddc  12293  imasaddfnlemg  13581  sraring  14726  distop  15079  cnptoprest  15233  upgr1edc  16245  uspgr1edc  16364  pw1ndom3lem  16902  pwle2  16911  pw1nct  16916