Theorem sseqtrrdi 3232

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 3231	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  4515  iotanul  5234  iotass  5236  tfrlem9  6377  tfrlemibfn  6386  tfrlemiubacc  6388  tfrlemi14d  6391  tfr1onlemssrecs  6397  tfr1onlemres  6407  tfrcllemres  6420  exmidfodomrlemr  7269  exmidfodomrlemrALT  7270  uznnssnn  9651  shftfvalg  10983  shftfval  10986  clim2prod  11704  reldvdsrsrg  13648  dvdsrvald  13649  dvdsrex  13654  eltopss  14245  difopn  14344  tgrest  14405  txuni2  14492  tgioo  14790  plycoeid3  14993