Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  celaront Structured version   Visualization version   GIF version

Theorem celaront 2556
 Description: "Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
celaront.maj 𝑥(𝜑 → ¬ 𝜓)
celaront.min 𝑥(𝜒𝜑)
celaront.e 𝑥𝜒
Assertion
Ref Expression
celaront 𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem celaront
StepHypRef Expression
1 celaront.maj . 2 𝑥(𝜑 → ¬ 𝜓)
2 celaront.min . 2 𝑥(𝜒𝜑)
3 celaront.e . 2 𝑥𝜒
41, 2, 3barbari 2555 1 𝑥(𝜒 ∧ ¬ 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator