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Theorem barbari 2667
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 https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Hypotheses
Ref Expression
barbari.maj 𝑥(𝜑𝜓)
barbari.min 𝑥(𝜒𝜑)
barbari.e 𝑥𝜒
Assertion
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 2661 . 2 𝑥(𝜒𝜓)
51, 4barbarilem 2666 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777
This theorem is referenced by:  celaront  2669  bamalip  2690
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