Theorem sseqtrrdi 3276

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

Hypotheses

Ref	Expression
sseqtrrdi.1	|- ( ph -> A C_ B )
sseqtrrdi.2	|- C = B

Assertion

Ref	Expression
sseqtrrdi	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrdi

Step	Hyp	Ref	Expression
1		sseqtrrdi.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrdi.2	. . 3 |- C = B
3	2	eqcomi 2235	. 2 |- B = C
4	1, 3	sseqtrdi 3275	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1397   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  iunpw  4577  iotanul  5302  iotass  5304  tfrlem9  6484  tfrlemibfn  6493  tfrlemiubacc  6495  tfrlemi14d  6498  tfr1onlemssrecs  6504  tfr1onlemres  6514  tfrcllemres  6527  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  uznnssnn  9810  pfxccatpfx2  11317  shftfvalg  11378  shftfval  11381  clim2prod  12099  dvdsrvald  14106  dvdsrex  14111  eltopss  14732  difopn  14831  tgrest  14892  txuni2  14979  tgioo  15277  plycoeid3  15480