Theorem sseqtrrdi 3289

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

Hypotheses

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

Assertion

Ref	Expression
sseqtrrdi	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sseqtrrdi

Step	Hyp	Ref	Expression
1		sseqtrrdi.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseqtrrdi.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2238	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3288	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1398   ⊆ wss 3213

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3219  df-ss 3226

This theorem is referenced by:  iunpw  4603  iotanul  5330  iotass  5332  tfrlem9  6552  tfrlemibfn  6561  tfrlemiubacc  6563  tfrlemi14d  6566  tfr1onlemssrecs  6572  tfr1onlemres  6582  tfrcllemres  6595  exmidfodomrlemr  7507  exmidfodomrlemrALT  7508  uznnssnn  9915  pfxccatpfx2  11437  shftfvalg  11511  shftfval  11514  clim2prod  12233  dvdsrvald  14260  dvdsrex  14265  eltopss  14923  difopn  15022  tgrest  15083  txuni2  15170  tgioo  15468  plycoeid3  15671