Theorem sseqtrrdi 3219

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 2193	. 2 |- B = C
4	1, 3	sseqtrdi 3218	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  iunpw  4495  iotanul  5208  iotass  5210  tfrlem9  6338  tfrlemibfn  6347  tfrlemiubacc  6349  tfrlemi14d  6352  tfr1onlemssrecs  6358  tfr1onlemres  6368  tfrcllemres  6381  exmidfodomrlemr  7220  exmidfodomrlemrALT  7221  uznnssnn  9596  shftfvalg  10846  shftfval  10849  clim2prod  11566  reldvdsrsrg  13409  dvdsrvald  13410  dvdsrex  13415  eltopss  13912  difopn  14011  tgrest  14072  txuni2  14159  tgioo  14449