Theorem syl6sseqr 3096

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

Hypotheses

Ref	Expression
syl6ssr.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
syl6ssr.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
syl6sseqr	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl6sseqr

Step	Hyp	Ref	Expression
1		syl6ssr.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		syl6ssr.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2104	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	syl6sseq 3095	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1299   ⊆ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034

This theorem is referenced by:  iunpw  4339  iotanul  5039  iotass  5041  tfrlem9  6146  tfrlemibfn  6155  tfrlemiubacc  6157  tfrlemi14d  6160  tfr1onlemssrecs  6166  tfr1onlemres  6176  tfrcllemres  6189  exmidfodomrlemr  6967  exmidfodomrlemrALT  6968  uznnssnn  9222  shftfvalg  10431  shftfval  10434  eltopss  11958  difopn  12059  tgrest  12120  txuni2  12206  tgioo  12465