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Theorem dimatis 2731
Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2714 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
dimatis.maj 𝑥(𝜑𝜓)
dimatis.min 𝑥(𝜓𝜒)
Assertion
Ref Expression
dimatis 𝑥(𝜒𝜑)

Proof of Theorem dimatis
StepHypRef Expression
1 dimatis.maj . 2 𝑥(𝜑𝜓)
2 dimatis.min . . . . 5 𝑥(𝜓𝜒)
32spi 2208 . . . 4 (𝜓𝜒)
43adantl 467 . . 3 ((𝜑𝜓) → 𝜒)
5 simpl 468 . . 3 ((𝜑𝜓) → 𝜑)
64, 5jca 501 . 2 ((𝜑𝜓) → (𝜒𝜑))
71, 6eximii 1912 1 𝑥(𝜒𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by: (None)
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