Theorem sseqtrrdi 3216

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 2191	. 2 |- B = C
4	1, 3	sseqtrdi 3215	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1363   C_ wss 3141

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154

This theorem is referenced by:  iunpw  4492  iotanul  5205  iotass  5207  tfrlem9  6333  tfrlemibfn  6342  tfrlemiubacc  6344  tfrlemi14d  6347  tfr1onlemssrecs  6353  tfr1onlemres  6363  tfrcllemres  6376  exmidfodomrlemr  7214  exmidfodomrlemrALT  7215  uznnssnn  9590  shftfvalg  10840  shftfval  10843  clim2prod  11560  reldvdsrsrg  13324  dvdsrvald  13325  dvdsrex  13330  eltopss  13749  difopn  13848  tgrest  13909  txuni2  13996  tgioo  14286