Theorem sseqtrrdi 3191

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 2169	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3190	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1343   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  iunpw  4458  iotanul  5168  iotass  5170  tfrlem9  6287  tfrlemibfn  6296  tfrlemiubacc  6298  tfrlemi14d  6301  tfr1onlemssrecs  6307  tfr1onlemres  6317  tfrcllemres  6330  exmidfodomrlemr  7158  exmidfodomrlemrALT  7159  uznnssnn  9515  shftfvalg  10760  shftfval  10763  clim2prod  11480  eltopss  12647  difopn  12748  tgrest  12809  txuni2  12896  tgioo  13186