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Theorem sylan9ss 3995
Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Hypotheses
Ref Expression
sylan9ss.1 (𝜑𝐴𝐵)
sylan9ss.2 (𝜓𝐵𝐶)
Assertion
Ref Expression
sylan9ss ((𝜑𝜓) → 𝐴𝐶)

Proof of Theorem sylan9ss
StepHypRef Expression
1 sylan9ss.1 . 2 (𝜑𝐴𝐵)
2 sylan9ss.2 . 2 (𝜓𝐵𝐶)
3 sstr 3990 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2an 595 1 ((𝜑𝜓) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965
This theorem is referenced by:  sylan9ssr  3996  psstr  4104  unss12  4182  ss2in  4236  ssdisj  4459  relrelss  6272  funssxp  6746  axdc3lem  10449  tskuni  10782  rtrclreclem4  15013  tsmsxp  23880  shslubi  30906  chlej12i  30996  insiga  33434  fnetr  35540  pcl0bN  39098  brtrclfv2  42781
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