Theorem sseqtrrdi 3291

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 2238	. 2 |- B = C
4	1, 3	sseqtrdi 3290	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3214

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 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  iunpw  4606  iotanul  5333  iotass  5335  tfrlem9  6563  tfrlemibfn  6572  tfrlemiubacc  6574  tfrlemi14d  6577  tfr1onlemssrecs  6583  tfr1onlemres  6593  tfrcllemres  6606  exmidfodomrlemr  7518  exmidfodomrlemrALT  7519  uznnssnn  9927  pfxccatpfx2  11454  shftfvalg  11528  shftfval  11531  clim2prod  12250  dvdsrvald  14338  dvdsrex  14343  eltopss  15000  difopn  15099  tgrest  15160  txuni2  15247  tgioo  15545  plycoeid3  15748