Theorem sseqtrrdi 3196

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 2174	. 2 |- B = C
4	1, 3	sseqtrdi 3195	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1348   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  iunpw  4463  iotanul  5173  iotass  5175  tfrlem9  6296  tfrlemibfn  6305  tfrlemiubacc  6307  tfrlemi14d  6310  tfr1onlemssrecs  6316  tfr1onlemres  6326  tfrcllemres  6339  exmidfodomrlemr  7172  exmidfodomrlemrALT  7173  uznnssnn  9529  shftfvalg  10775  shftfval  10778  clim2prod  11495  eltopss  12766  difopn  12867  tgrest  12928  txuni2  13015  tgioo  13305