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Theorem fresison 2571
 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(𝜓𝜒)
Assertion
Ref Expression
fresison 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison
StepHypRef Expression
1 fresison.min . 2 𝑥(𝜓𝜒)
2 simpr 476 . . 3 ((𝜓𝜒) → 𝜒)
3 fresison.maj . . . . . 6 𝑥(𝜑 → ¬ 𝜓)
43spi 2042 . . . . 5 (𝜑 → ¬ 𝜓)
54con2i 133 . . . 4 (𝜓 → ¬ 𝜑)
65adantr 480 . . 3 ((𝜓𝜒) → ¬ 𝜑)
72, 6jca 553 . 2 ((𝜓𝜒) → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1754 1 𝑥(𝜒 ∧ ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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