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  6386  tfrlemibfn  6395  tfrlemiubacc  6397  tfrlemi14d  6400  tfr1onlemssrecs  6406  tfr1onlemres  6416  tfrcllemres  6429  exmidfodomrlemr  7281  exmidfodomrlemrALT  7282  uznnssnn  9668  shftfvalg  11000  shftfval  11003  clim2prod  11721  reldvdsrsrg  13724  dvdsrvald  13725  dvdsrex  13730  eltopss  14329  difopn  14428  tgrest  14489  txuni2  14576  tgioo  14874  plycoeid3  15077