Theorem cesaro

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps , all ch is ps , and ch exist, therefore some ch is not ph . In Aristotelian notation, EAO-2: PeM and SaM therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	cesaro.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
	cesaro.min	⊢ ∀ 𝑥 ( 𝜒 → 𝜓 )
	cesaro.e	⊢ ∃ 𝑥 𝜒
Assertion	cesaro	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		cesaro.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
2		cesaro.min	⊢ ∀ 𝑥 ( 𝜒 → 𝜓 )
3		cesaro.e	⊢ ∃ 𝑥 𝜒
4	1 2	cesare	⊢ ∀ 𝑥 ( 𝜒 → ¬ 𝜑 )
5	3 4	barbarilem	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )