Theorem syl6sseq 3045

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 3024	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	sylib 120	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1284   ⊆ wss 2973

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  syl6sseqr  3046  onintonm  4261  relrelss  4864  iotanul  4902  foimacnv  5164  cauappcvgprlemladdru  6846