Theorem sseqtrrdi 3297

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 2242	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3296	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1402   ⊆ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  iunpw  4624  iotanul  5351  iotass  5353  tfrlem9  6584  tfrlemibfn  6593  tfrlemiubacc  6595  tfrlemi14d  6598  tfr1onlemssrecs  6604  tfr1onlemres  6614  tfrcllemres  6627  exmidfodomrlemr  7548  exmidfodomrlemrALT  7549  uznnssnn  9960  pfxccatpfx2  11492  shftfvalg  11566  shftfval  11569  clim2prod  12289  dvdsrvald  14383  dvdsrex  14388  eltopss  15093  difopn  15192  tgrest  15253  txuni2  15340  tgioo  15638  plycoeid3  15841