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Theorem fesapo 2323
 Description: "Fesapo", one of the syllogisms of Aristotelian logic. No φ is ψ, all ψ is χ, and ψ exist, therefore some χ is not φ. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fesapo.maj x(φ → ¬ ψ)
fesapo.min x(ψχ)
fesapo.e xψ
Assertion
Ref Expression
fesapo x(χ ¬ φ)

Proof of Theorem fesapo
StepHypRef Expression
1 fesapo.e . 2 xψ
2 fesapo.min . . . . 5 x(ψχ)
32spi 1753 . . . 4 (ψχ)
4 fesapo.maj . . . . . 6 x(φ → ¬ ψ)
54spi 1753 . . . . 5 (φ → ¬ ψ)
65con2i 112 . . . 4 (ψ → ¬ φ)
73, 6jca 518 . . 3 (ψ → (χ ¬ φ))
87eximi 1576 . 2 (xψx(χ ¬ φ))
91, 8ax-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)
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