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Theorem cesare 2557
 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 2552. (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 2042 . . 3 (𝜑 → ¬ 𝜓)
3 cesare.min . . . 4 𝑥(𝜒𝜓)
43spi 2042 . . 3 (𝜒𝜓)
52, 4nsyl3 132 . 2 (𝜒 → ¬ 𝜑)
65ax-gen 1713 1 𝑥(𝜒 → ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 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-ex 1696 This theorem is referenced by: (None)
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