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  4570  iotanul  5293  iotass  5295  tfrlem9  6463  tfrlemibfn  6472  tfrlemiubacc  6474  tfrlemi14d  6477  tfr1onlemssrecs  6483  tfr1onlemres  6493  tfrcllemres  6506  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  uznnssnn  9768  pfxccatpfx2  11264  shftfvalg  11324  shftfval  11327  clim2prod  12045  reldvdsrsrg  14050  dvdsrvald  14051  dvdsrex  14056  eltopss  14677  difopn  14776  tgrest  14837  txuni2  14924  tgioo  15222  plycoeid3  15425