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Theorem fresison 2321
 Description: "Fresison", one of the syllogisms of Aristotelian logic. No φ is ψ (PeM), and some ψ is χ (MiS), therefore some χ is not φ (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fresison.maj x(φ → ¬ ψ)
fresison.min x(ψ χ)
Assertion
Ref Expression
fresison x(χ ¬ φ)

Proof of Theorem fresison
StepHypRef Expression
1 fresison.min . 2 x(ψ χ)
2 simpr 447 . . . 4 ((ψ χ) → χ)
3 fresison.maj . . . . . . 7 x(φ → ¬ ψ)
43spi 1753 . . . . . 6 (φ → ¬ ψ)
54con2i 112 . . . . 5 (ψ → ¬ φ)
65adantr 451 . . . 4 ((ψ χ) → ¬ φ)
72, 6jca 518 . . 3 ((ψ χ) → (χ ¬ φ))
87eximi 1576 . 2 (x(ψ χ) → x(χ ¬ φ))
91, 8ax-mp 5 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-mp 5  ax-1 6  ax-2 7  ax-3 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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