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Theorem darii 2303
 Description: "Darii", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is φ, therefore some χ is ψ. (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
darii.maj x(φψ)
darii.min x(χ φ)
Assertion
Ref Expression
darii x(χ ψ)

Proof of Theorem darii
StepHypRef Expression
1 darii.min . 2 x(χ φ)
2 darii.maj . . . . 5 x(φψ)
32spi 1753 . . . 4 (φψ)
43anim2i 552 . . 3 ((χ φ) → (χ ψ))
54eximi 1576 . 2 (x(χ φ) → x(χ ψ))
61, 5ax-mp 8 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  ferio  2304
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