Theorem sseqtrrdi 3241

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 2208	. 2 |- B = C
4	1, 3	sseqtrdi 3240	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1372   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  iunpw  4526  iotanul  5246  iotass  5248  tfrlem9  6404  tfrlemibfn  6413  tfrlemiubacc  6415  tfrlemi14d  6418  tfr1onlemssrecs  6424  tfr1onlemres  6434  tfrcllemres  6447  exmidfodomrlemr  7309  exmidfodomrlemrALT  7310  uznnssnn  9697  shftfvalg  11100  shftfval  11103  clim2prod  11821  reldvdsrsrg  13825  dvdsrvald  13826  dvdsrex  13831  eltopss  14452  difopn  14551  tgrest  14612  txuni2  14699  tgioo  14997  plycoeid3  15200