Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  barbari Structured version   Unicode version

Theorem barbari 2405
 Description: "Barbari", one of the syllogisms of Aristotelian logic. All is , all is , and some exist, therefore some is . (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
barbari.min
barbari.e
Assertion
Ref Expression
barbari

Proof of Theorem barbari
StepHypRef Expression
1 barbari.e . 2
2 barbari.maj . . . . 5
3 barbari.min . . . . 5
42, 3barbara 2401 . . . 4
54spi 1808 . . 3
65ancli 551 . 2
71, 6eximii 1632 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369  wal 1372  wex 1591 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592 This theorem is referenced by:  celaront  2406
 Copyright terms: Public domain W3C validator