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Theorem fesapo 2753
 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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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.maj . . 3 𝑥(𝜑 → ¬ 𝜓)
2 con2 137 . . . 4 ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
32alimi 1813 . . 3 (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥(𝜓 → ¬ 𝜑))
41, 3ax-mp 5 . 2 𝑥(𝜓 → ¬ 𝜑)
5 fesapo.min . 2 𝑥(𝜓𝜒)
6 fesapo.e . 2 𝑥𝜓
74, 5, 6felapton 2748 1 𝑥(𝜒 ∧ ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by: (None)
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