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Theorem darii 2100
 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 1517 . . 3 (𝜑𝜓)
43anim2i 340 . 2 ((𝜒𝜑) → (𝜒𝜓))
51, 4eximii 1582 1 𝑥(𝜒𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  ferio  2101
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