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

Theorem festino 2309
 Description: "Festino", one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is ψ, therefore some χ is not φ. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
Hypotheses
Ref Expression
festino.maj x(φ → ¬ ψ)
festino.min x(χ ψ)
Assertion
Ref Expression
festino x(χ ¬ φ)

Proof of Theorem festino
StepHypRef Expression
1 festino.min . 2 x(χ ψ)
2 festino.maj . . . . . 6 x(φ → ¬ ψ)
32spi 1753 . . . . 5 (φ → ¬ ψ)
43con2i 112 . . . 4 (ψ → ¬ φ)
54anim2i 552 . . 3 ((χ ψ) → (χ ¬ φ))
65eximi 1576 . 2 (x(χ ψ) → x(χ ¬ φ))
71, 6ax-mp 8 1 x(χ ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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: (None)
 Copyright terms: Public domain W3C validator