Theorem syl6sseq 3056

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

Hypotheses

Ref	Expression
syl6sseq.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
syl6sseq.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
syl6sseq	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl6sseq

Step	Hyp	Ref	Expression
1		syl6sseq.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		syl6sseq.2	. . 3 ⊢ 𝐵 = 𝐶
3	2	sseq2i 3035	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	sylib 120	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1285   ⊆ wss 2984

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2990  df-ss 2997

This theorem is referenced by:  syl6sseqr  3057  onintonm  4297  relrelss  4911  iotanul  4949  foimacnv  5219  cauappcvgprlemladdru  7118