Theorem sseqtrrdi 3274

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 2233	. 2 |- B = C
4	1, 3	sseqtrdi 3273	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1395   C_ wss 3198

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3204  df-ss 3211

This theorem is referenced by:  iunpw  4575  iotanul  5300  iotass  5302  tfrlem9  6480  tfrlemibfn  6489  tfrlemiubacc  6491  tfrlemi14d  6494  tfr1onlemssrecs  6500  tfr1onlemres  6510  tfrcllemres  6523  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  uznnssnn  9801  pfxccatpfx2  11308  shftfvalg  11369  shftfval  11372  clim2prod  12090  dvdsrvald  14097  dvdsrex  14102  eltopss  14723  difopn  14822  tgrest  14883  txuni2  14970  tgioo  15268  plycoeid3  15471