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Theorem sylan9ss 2986
 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 2981 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2an 277 1 ((𝜑𝜓) → 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ⊆ wss 2945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959 This theorem is referenced by:  sylan9ssr  2987  psstr  3077  sspsstr  3078  psssstr  3079  unss12  3143  ss2in  3192  relrelss  4872  funssxp  5088
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