Theorem sseqtrrdi 3273

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 2233	. 2 |- B = C
4	1, 3	sseqtrdi 3272	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3197

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  iunpw  4571  iotanul  5294  iotass  5296  tfrlem9  6471  tfrlemibfn  6480  tfrlemiubacc  6482  tfrlemi14d  6485  tfr1onlemssrecs  6491  tfr1onlemres  6501  tfrcllemres  6514  exmidfodomrlemr  7391  exmidfodomrlemrALT  7392  uznnssnn  9784  pfxccatpfx2  11284  shftfvalg  11344  shftfval  11347  clim2prod  12065  dvdsrvald  14072  dvdsrex  14077  eltopss  14698  difopn  14797  tgrest  14858  txuni2  14945  tgioo  15243  plycoeid3  15446