Theorem sseqtrrdi 3297

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 2242	. 2 |- B = C
4	1, 3	sseqtrdi 3296	1 |- ( ph -> A C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  iunpw  4626  iotanul  5353  iotass  5355  tfrlem9  6590  tfrlemibfn  6599  tfrlemiubacc  6601  tfrlemi14d  6604  tfr1onlemssrecs  6610  tfr1onlemres  6620  tfrcllemres  6633  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555  uznnssnn  9977  pfxccatpfx2  11509  shftfvalg  11583  shftfval  11586  clim2prod  12306  dvdsrvald  14400  dvdsrex  14405  eltopss  15110  difopn  15209  tgrest  15270  txuni2  15357  tgioo  15655  plycoeid3  15858