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Theorem dimatis 2776
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 2753 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
Ref Expression
dimatis.maj 𝑥(𝜑𝜓)
dimatis.min 𝑥(𝜓𝜒)
Ref Expression
dimatis 𝑥(𝜒𝜑)

Proof of Theorem dimatis
StepHypRef Expression
1 dimatis.min . . 3 𝑥(𝜓𝜒)
2 dimatis.maj . . 3 𝑥(𝜑𝜓)
31, 2darii 2753 . 2 𝑥(𝜑𝜒)
4 exancom 1854 . 2 (∃𝑥(𝜑𝜒) ↔ ∃𝑥(𝜒𝜑))
53, 4mpbi 231 1 𝑥(𝜒𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774
This theorem is referenced by: (None)
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