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Theorem sseqtrdi 3176
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtrdi.1  |-  ( ph  ->  A  C_  B )
sseqtrdi.2  |-  B  =  C
Assertion
Ref Expression
sseqtrdi  |-  ( ph  ->  A  C_  C )

Proof of Theorem sseqtrdi
StepHypRef Expression
1 sseqtrdi.1 . 2  |-  ( ph  ->  A  C_  B )
2 sseqtrdi.2 . . 3  |-  B  =  C
32sseq2i 3155 . 2  |-  ( A 
C_  B  <->  A  C_  C
)
41, 3sylib 121 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    C_ wss 3102
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115
This theorem is referenced by:  sseqtrrdi  3177  onintonm  4477  relrelss  5113  iotanul  5151  foimacnv  5433  cauappcvgprlemladdru  7577  zsumdc  11285  fsum3cvg3  11297  zproddc  11480  nninfdcex  11843  distop  12527  cnptoprest  12681  pwle2  13612  pw1nct  13617
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