Theorem sseqtrrdi 3243

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 2210	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3242	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1373   ⊆ wss 3167

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3173  df-ss 3180

This theorem is referenced by:  iunpw  4531  iotanul  5252  iotass  5254  tfrlem9  6412  tfrlemibfn  6421  tfrlemiubacc  6423  tfrlemi14d  6426  tfr1onlemssrecs  6432  tfr1onlemres  6442  tfrcllemres  6455  exmidfodomrlemr  7317  exmidfodomrlemrALT  7318  uznnssnn  9705  shftfvalg  11173  shftfval  11176  clim2prod  11894  reldvdsrsrg  13898  dvdsrvald  13899  dvdsrex  13904  eltopss  14525  difopn  14624  tgrest  14685  txuni2  14772  tgioo  15070  plycoeid3  15273