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Theorem barbari 2044
 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 2040 . . . 4 𝑥(𝜒𝜓)
54spi 1470 . . 3 (𝜒𝜓)
65ancli 316 . 2 (𝜒 → (𝜒𝜓))
71, 6eximii 1534 1 𝑥(𝜒𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1283  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468 This theorem depends on definitions:  df-bi 115 This theorem is referenced by:  celaront  2045
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