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Theorem sylan9ss 3110
 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 3105 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2an 287 1 ((𝜑𝜓) → 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ⊆ wss 3071 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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084 This theorem is referenced by:  sylan9ssr  3111  unss12  3248  ss2in  3304  relrelss  5068  funssxp  5295
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