Theorem sseqtrrdi 3297

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 2242	. 2 |- B = C
4	1, 3	sseqtrdi 3296	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  iunpw  4621  iotanul  5348  iotass  5350  tfrlem9  6580  tfrlemibfn  6589  tfrlemiubacc  6591  tfrlemi14d  6594  tfr1onlemssrecs  6600  tfr1onlemres  6610  tfrcllemres  6623  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  uznnssnn  9956  pfxccatpfx2  11487  shftfvalg  11561  shftfval  11564  clim2prod  12284  dvdsrvald  14373  dvdsrex  14378  eltopss  15033  difopn  15132  tgrest  15193  txuni2  15280  tgioo  15578  plycoeid3  15781