Theorem sseqtrrdi 3146

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 2143	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	sseqtrdi 3145	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1331   ⊆ wss 3071

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  iunpw  4401  iotanul  5103  iotass  5105  tfrlem9  6216  tfrlemibfn  6225  tfrlemiubacc  6227  tfrlemi14d  6230  tfr1onlemssrecs  6236  tfr1onlemres  6246  tfrcllemres  6259  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  uznnssnn  9372  shftfvalg  10590  shftfval  10593  clim2prod  11308  eltopss  12176  difopn  12277  tgrest  12338  txuni2  12425  tgioo  12715