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Theorem cesare 2556
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 2551. (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 2041 . . 3 (𝜑 → ¬ 𝜓)
3 cesare.min . . . 4 𝑥(𝜒𝜓)
43spi 2041 . . 3 (𝜒𝜓)
52, 4nsyl3 131 . 2 (𝜒 → ¬ 𝜑)
65ax-gen 1712 1 𝑥(𝜒 → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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