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Theorem fresison 2777
 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.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
fresison.maj 𝑥(𝜑 → ¬ 𝜓)
fresison.min 𝑥(𝜓𝜒)
Assertion
Ref Expression
fresison 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison
StepHypRef Expression
1 fresison.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 fresison.min . . 3 𝑥(𝜓𝜒)
3 exancom 1854 . . 3 (∃𝑥(𝜓𝜒) ↔ ∃𝑥(𝜒𝜓))
42, 3mpbi 231 . 2 𝑥(𝜒𝜓)
51, 4festino 2762 1 𝑥(𝜒 ∧ ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1528  ∃wex 1773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774 This theorem is referenced by: (None)
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