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Theorem celarent 2302
 Description: "Celarent", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is φ, therefore no χ is ψ. (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
celarent.maj x(φ → ¬ ψ)
celarent.min x(χφ)
Assertion
Ref Expression
celarent x(χ → ¬ ψ)

Proof of Theorem celarent
StepHypRef Expression
1 celarent.maj . 2 x(φ → ¬ ψ)
2 celarent.min . 2 x(χφ)
31, 2barbara 2301 1 x(χ → ¬ ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360 This theorem is referenced by: (None)
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