Theorem sseqtrrdi 3219

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 2193	. 2 |- B = C
4	1, 3	sseqtrdi 3218	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  iunpw  4498  iotanul  5211  iotass  5213  tfrlem9  6343  tfrlemibfn  6352  tfrlemiubacc  6354  tfrlemi14d  6357  tfr1onlemssrecs  6363  tfr1onlemres  6373  tfrcllemres  6386  exmidfodomrlemr  7230  exmidfodomrlemrALT  7231  uznnssnn  9606  shftfvalg  10858  shftfval  10861  clim2prod  11578  reldvdsrsrg  13439  dvdsrvald  13440  dvdsrex  13445  eltopss  13961  difopn  14060  tgrest  14121  txuni2  14208  tgioo  14498