Theorem sseqtrrdi 3276

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 3275	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1397   C_ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  iunpw  4577  iotanul  5302  iotass  5304  tfrlem9  6485  tfrlemibfn  6494  tfrlemiubacc  6496  tfrlemi14d  6499  tfr1onlemssrecs  6505  tfr1onlemres  6515  tfrcllemres  6528  exmidfodomrlemr  7413  exmidfodomrlemrALT  7414  uznnssnn  9811  pfxccatpfx2  11318  shftfvalg  11379  shftfval  11382  clim2prod  12101  dvdsrvald  14109  dvdsrex  14114  eltopss  14735  difopn  14834  tgrest  14895  txuni2  14982  tgioo  15280  plycoeid3  15483