Theorem sseqtrrdi 3297

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 2242	. 2 |- B = C
4	1, 3	sseqtrdi 3296	1 |- ( ph -> A C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on 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 proof 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 used by:  iunpw  4628  iotanul  5356  iotass  5358  tfrlem9  6595  tfrlemibfn  6604  tfrlemiubacc  6606  tfrlemi14d  6609  tfr1onlemssrecs  6615  tfr1onlemres  6625  tfrcllemres  6638  exmidfodomrlemr  7560  exmidfodomrlemrALT  7561  uznnssnn  9992  pfxccatpfx2  11530  shftfvalg  11604  shftfval  11607  clim2prod  12330  dvdsrvald  14508  dvdsrex  14513  eltopss  15252  difopn  15351  tgrest  15412  txuni2  15499  tgioo  15797  plycoeid3  16000