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

Theorem barbari 2079
Description: "Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
Hypotheses
Ref Expression
barbari.maj 𝑥(𝜑𝜓)
barbari.min 𝑥(𝜒𝜑)
barbari.e 𝑥𝜒
Assertion
Ref Expression
barbari 𝑥(𝜒𝜓)

Proof of Theorem barbari
StepHypRef Expression
1 barbari.e . 2 𝑥𝜒
2 barbari.maj . . . . 5 𝑥(𝜑𝜓)
3 barbari.min . . . . 5 𝑥(𝜒𝜑)
42, 3barbara 2075 . . . 4 𝑥(𝜒𝜓)
54spi 1501 . . 3 (𝜒𝜓)
65ancli 321 . 2 (𝜒 → (𝜒𝜓))
71, 6eximii 1566 1 𝑥(𝜒𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1314  wex 1453
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  celaront  2080
  Copyright terms: Public domain W3C validator