Theorem sseqtrrdi 3287

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 2236	. 2 |- B = C
4	1, 3	sseqtrdi 3286	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3211

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224

This theorem is referenced by:  iunpw  4601  iotanul  5328  iotass  5330  tfrlem9  6550  tfrlemibfn  6559  tfrlemiubacc  6561  tfrlemi14d  6564  tfr1onlemssrecs  6570  tfr1onlemres  6580  tfrcllemres  6593  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  uznnssnn  9909  pfxccatpfx2  11429  shftfvalg  11503  shftfval  11506  clim2prod  12225  dvdsrvald  14238  dvdsrex  14243  eltopss  14874  difopn  14973  tgrest  15034  txuni2  15121  tgioo  15419  plycoeid3  15622