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

Theorem ferio 2044
Description: "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
ferio.maj 𝑥(𝜑 → ¬ 𝜓)
ferio.min 𝑥(𝜒𝜑)
Assertion
Ref Expression
ferio 𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem ferio
StepHypRef Expression
1 ferio.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 ferio.min . 2 𝑥(𝜒𝜑)
31, 2darii 2043 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator