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Theorem dimatis 2611
Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2594 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
dimatis.maj 𝑥(𝜑𝜓)
dimatis.min 𝑥(𝜓𝜒)
Assertion
Ref Expression
dimatis 𝑥(𝜒𝜑)

Proof of Theorem dimatis
StepHypRef Expression
1 dimatis.maj . 2 𝑥(𝜑𝜓)
2 dimatis.min . . . . 5 𝑥(𝜓𝜒)
32spi 2092 . . . 4 (𝜓𝜒)
43adantl 481 . . 3 ((𝜑𝜓) → 𝜒)
5 simpl 472 . . 3 ((𝜑𝜓) → 𝜑)
64, 5jca 553 . 2 ((𝜑𝜓) → (𝜒𝜑))
71, 6eximii 1804 1 𝑥(𝜒𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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