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Theorem darii 2564
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 2052 . . 3 (𝜑𝜓)
43anim2i 592 . 2 ((𝜒𝜑) → (𝜒𝜓))
51, 4eximii 1761 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  ferio  2565
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