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Theorem darapti 2318
 Description: "Darapti", one of the syllogisms of Aristotelian logic. All φ is ψ, all φ is χ, and some φ exist, therefore some χ is ψ. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
darapti.maj x(φψ)
darapti.min x(φχ)
darapti.e xφ
Assertion
Ref Expression
darapti x(χ ψ)

Proof of Theorem darapti
StepHypRef Expression
1 darapti.e . 2 xφ
2 darapti.min . . . . 5 x(φχ)
32spi 1753 . . . 4 (φχ)
4 darapti.maj . . . . 5 x(φψ)
54spi 1753 . . . 4 (φψ)
63, 5jca 518 . . 3 (φ → (χ ψ))
76eximi 1576 . 2 (xφx(χ ψ))
81, 7ax-mp 5 1 x(χ ψ)
 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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