ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sylan9ss GIF version

Theorem sylan9ss 3183
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 3178 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2an 289 1 ((𝜑𝜓) → 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157
This theorem is referenced by:  sylan9ssr  3184  unss12  3322  ss2in  3378  relrelss  5170  funssxp  5400
  Copyright terms: Public domain W3C validator