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  4628  iotanul  5356  iotass  5358  tfrlem9  6594  tfrlemibfn  6603  tfrlemiubacc  6605  tfrlemi14d  6608  tfr1onlemssrecs  6614  tfr1onlemres  6624  tfrcllemres  6637  exmidfodomrlemr  7560  exmidfodomrlemrALT  7561  uznnssnn  9993  pfxccatpfx2  11531  shftfvalg  11605  shftfval  11608  clim2prod  12331  dvdsrvald  14509  dvdsrex  14514  eltopss  15235  difopn  15334  tgrest  15395  txuni2  15482  tgioo  15780  plycoeid3  15983