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Theorem sseqtrdi 3195
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 3174 . 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 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:  sseqtrrdi  3196  onintonm  4501  relrelss  5137  iotanul  5175  foimacnv  5460  cauappcvgprlemladdru  7618  zsumdc  11347  fsum3cvg3  11359  zproddc  11542  nninfdcex  11908  zsupssdc  11909  distop  12879  cnptoprest  13033  pwle2  14031  pw1nct  14036
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