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Theorem dimatis 2320
 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 2303 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
dimatis.maj x(φ ψ)
dimatis.min x(ψχ)
Assertion
Ref Expression
dimatis x(χ φ)

Proof of Theorem dimatis
StepHypRef Expression
1 dimatis.maj . 2 x(φ ψ)
2 dimatis.min . . . . . 6 x(ψχ)
32spi 1753 . . . . 5 (ψχ)
43adantl 452 . . . 4 ((φ ψ) → χ)
5 simpl 443 . . . 4 ((φ ψ) → φ)
64, 5jca 518 . . 3 ((φ ψ) → (χ φ))
76eximi 1576 . 2 (x(φ ψ) → x(χ φ))
81, 7ax-mp 5 1 x(χ φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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