ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dimatis GIF version

Theorem dimatis 2060
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 2043 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 1470 . . . 4 (𝜓𝜒)
43adantl 271 . . 3 ((𝜑𝜓) → 𝜒)
5 simpl 107 . . 3 ((𝜑𝜓) → 𝜑)
64, 5jca 300 . 2 ((𝜑𝜓) → (𝜒𝜑))
71, 6eximii 1534 1 𝑥(𝜒𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator