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Theorem festino 2675
Description: "Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
festino.maj 𝑥(𝜑 → ¬ 𝜓)
festino.min 𝑥(𝜒𝜓)
Assertion
Ref Expression
festino 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem festino
StepHypRef Expression
1 festino.maj . . 3 𝑥(𝜑 → ¬ 𝜓)
2 con2 137 . . . . 5 ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
32anim2d 615 . . . 4 ((𝜑 → ¬ 𝜓) → ((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑)))
43alimi 1819 . . 3 (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑)))
51, 4ax-mp 5 . 2 𝑥((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑))
6 festino.min . 2 𝑥(𝜒𝜓)
7 exim 1841 . 2 (∀𝑥((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑)))
85, 6, 7mp2 9 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1541  wex 1787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788
This theorem is referenced by:  fresison  2690
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