New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  barbari GIF version

Theorem barbari 2305
 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(χ ψ)

Proof of Theorem barbari
StepHypRef Expression
1 barbari.e . 2 xχ
2 barbari.maj . . . . . 6 x(φψ)
3 barbari.min . . . . . 6 x(χφ)
42, 3barbara 2301 . . . . 5 x(χψ)
54spi 1753 . . . 4 (χψ)
65ancli 534 . . 3 (χ → (χ ψ))
76eximi 1576 . 2 (xχx(χ ψ))
81, 7ax-mp 8 1 x(χ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by:  celaront  2306
 Copyright terms: Public domain W3C validator