Theorem sseqtrrdi 3151

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 2144	. 2 |- B = C
4	1, 3	sseqtrdi 3150	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1332   C_ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3082  df-ss 3089

This theorem is referenced by:  iunpw  4409  iotanul  5111  iotass  5113  tfrlem9  6224  tfrlemibfn  6233  tfrlemiubacc  6235  tfrlemi14d  6238  tfr1onlemssrecs  6244  tfr1onlemres  6254  tfrcllemres  6267  exmidfodomrlemr  7075  exmidfodomrlemrALT  7076  uznnssnn  9399  shftfvalg  10622  shftfval  10625  clim2prod  11340  eltopss  12215  difopn  12316  tgrest  12377  txuni2  12464  tgioo  12754