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Theorem cesare 2307
 Description: "Cesare", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is ψ, therefore no χ is φ. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2302. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
Hypotheses
Ref Expression
cesare.maj x(φ → ¬ ψ)
cesare.min x(χψ)
Assertion
Ref Expression
cesare x(χ → ¬ φ)

Proof of Theorem cesare
StepHypRef Expression
1 cesare.maj . . . 4 x(φ → ¬ ψ)
21spi 1753 . . 3 (φ → ¬ ψ)
3 cesare.min . . . 4 x(χψ)
43spi 1753 . . 3 (χψ)
52, 4nsyl3 111 . 2 (χ → ¬ φ)
65ax-gen 1546 1 x(χ → ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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