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Theorem barbari 2752
Description: "Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. In Aristotelian notation, AAI-1: MaP and SaM therefore SiP. For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Ref Expression
barbari.maj 𝑥(𝜑𝜓)
barbari.min 𝑥(𝜒𝜑)
barbari.e 𝑥𝜒
Ref Expression
barbari 𝑥(𝜒𝜓)

Proof of Theorem barbari
StepHypRef Expression
1 barbari.e . 2 𝑥𝜒
2 barbari.maj . . 3 𝑥(𝜑𝜓)
3 barbari.min . . 3 𝑥(𝜒𝜑)
42, 3barbara 2746 . 2 𝑥(𝜒𝜓)
51, 4barbarilem 2751 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1654  wex 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879
This theorem is referenced by:  celaront  2754  bamalip  2786
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