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Theorem dimatis 2581
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 2564 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 2052 . . . 4 (𝜓𝜒)
43adantl 482 . . 3 ((𝜑𝜓) → 𝜒)
5 simpl 473 . . 3 ((𝜑𝜓) → 𝜑)
64, 5jca 554 . 2 ((𝜑𝜓) → (𝜒𝜑))
71, 6eximii 1761 1 𝑥(𝜒𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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