Theorem sseqtrrdi 3233

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 2200	. 2 |- B = C
4	1, 3	sseqtrdi 3232	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1364   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  iunpw  4516  iotanul  5235  iotass  5237  tfrlem9  6378  tfrlemibfn  6387  tfrlemiubacc  6389  tfrlemi14d  6392  tfr1onlemssrecs  6398  tfr1onlemres  6408  tfrcllemres  6421  exmidfodomrlemr  7271  exmidfodomrlemrALT  7272  uznnssnn  9653  shftfvalg  10985  shftfval  10988  clim2prod  11706  reldvdsrsrg  13658  dvdsrvald  13659  dvdsrex  13664  eltopss  14255  difopn  14354  tgrest  14415  txuni2  14502  tgioo  14800  plycoeid3  15003