Theorem syl6sseq 3072

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 3051	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)
4	1, 3	sylib 120	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1289   ⊆ wss 2999

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  syl6sseqr  3073  onintonm  4332  relrelss  4952  iotanul  4990  foimacnv  5265  cauappcvgprlemladdru  7205  zisum  10761  fsum3cvg3  10776