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Theorem sylan9ss 3757
 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 3752 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2an 495 1 ((𝜑𝜓) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ⊆ wss 3715 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729 This theorem is referenced by:  sylan9ssr  3758  psstr  3853  unss12  3928  ss2in  3983  ssdisj  4170  relrelss  5820  funssxp  6222  axdc3lem  9464  tskuni  9797  rtrclreclem4  14000  tsmsxp  22159  shslubi  28553  chlej12i  28643  insiga  30509  fnetr  32652  pcl0bN  35712  brtrclfv2  38521
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