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Theorem darii 2552
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(𝜒𝜑)
Assertion
Ref Expression
darii 𝑥(𝜒𝜓)

Proof of Theorem darii
StepHypRef Expression
1 darii.min . 2 𝑥(𝜒𝜑)
2 darii.maj . . . 4 𝑥(𝜑𝜓)
32spi 2040 . . 3 (𝜑𝜓)
43anim2i 590 . 2 ((𝜒𝜑) → (𝜒𝜓))
51, 4eximii 1753 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  ferio  2553
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