Theorem sseqtrdi 3273

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 3252	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	sylib 122	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1395   ⊆ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  sseqtrrdi  3274  onintonm  4613  relrelss  5261  iotanul  5300  foimacnv  5598  pw1m  7435  cauappcvgprlemladdru  7869  nninfdcex  10490  zsupssdc  10491  zsumdc  11938  fsum3cvg3  11950  zproddc  12133  imasaddfnlemg  13390  sraring  14456  distop  14802  cnptoprest  14956  upgr1edc  15965  uspgr1edc  16084  pw1ndom3lem  16538  pwle2  16549  pw1nct  16554