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

Theorem fresison 2061
Description: "Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fresison.maj 𝑥(𝜑 → ¬ 𝜓)
fresison.min 𝑥(𝜓𝜒)
Assertion
Ref Expression
fresison 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison
StepHypRef Expression
1 fresison.min . 2 𝑥(𝜓𝜒)
2 simpr 108 . . 3 ((𝜓𝜒) → 𝜒)
3 fresison.maj . . . . . 6 𝑥(𝜑 → ¬ 𝜓)
43spi 1470 . . . . 5 (𝜑 → ¬ 𝜓)
54con2i 590 . . . 4 (𝜓 → ¬ 𝜑)
65adantr 270 . . 3 ((𝜓𝜒) → ¬ 𝜑)
72, 6jca 300 . 2 ((𝜓𝜒) → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1534 1 𝑥(𝜒 ∧ ¬ 𝜑)
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-in1 577  ax-in2 578  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