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Theorem barbari 2550
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.) (Revised by David A. Wheeler, 30-Aug-2016.)
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 . . . . 5 𝑥(𝜑𝜓)
3 barbari.min . . . . 5 𝑥(𝜒𝜑)
42, 3barbara 2546 . . . 4 𝑥(𝜒𝜓)
54spi 2039 . . 3 (𝜒𝜓)
65ancli 571 . 2 (𝜒 → (𝜒𝜓))
71, 6eximii 1752 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  celaront  2551
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