Theorem sseqtrrdi 3277

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 2235	. 2 |- B = C
4	1, 3	sseqtrdi 3276	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  iunpw  4583  iotanul  5309  iotass  5311  tfrlem9  6528  tfrlemibfn  6537  tfrlemiubacc  6539  tfrlemi14d  6542  tfr1onlemssrecs  6548  tfr1onlemres  6558  tfrcllemres  6571  exmidfodomrlemr  7456  exmidfodomrlemrALT  7457  uznnssnn  9855  pfxccatpfx2  11367  shftfvalg  11441  shftfval  11444  clim2prod  12163  dvdsrvald  14171  dvdsrex  14176  eltopss  14803  difopn  14902  tgrest  14963  txuni2  15050  tgioo  15348  plycoeid3  15551