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Theorem dimatis 2768
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 2745 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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.min . . 3 𝑥(𝜓𝜒)
2 dimatis.maj . . 3 𝑥(𝜑𝜓)
31, 2darii 2745 . 2 𝑥(𝜑𝜒)
4 exancom 1852 . 2 (∃𝑥(𝜑𝜒) ↔ ∃𝑥(𝜒𝜑))
53, 4mpbi 231 1 𝑥(𝜒𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
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