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Theorem cesare 2752
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 2744. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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 . . 3 𝑥(𝜑 → ¬ 𝜓)
2 con2 137 . . . 4 ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
32alimi 1803 . . 3 (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥(𝜓 → ¬ 𝜑))
41, 3ax-mp 5 . 2 𝑥(𝜓 → ¬ 𝜑)
5 cesare.min . 2 𝑥(𝜒𝜓)
64, 5celarent 2744 1 𝑥(𝜒 → ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397
This theorem is referenced by:  cesaro  2758
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