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Metamath Proof Explorer

Theorem cesare

Description: "Cesare", one of the syllogisms of Aristotelian logic. No ph is ps , and all ch is ps , therefore no ch is ph . In Aristotelian notation, EAE-2: PeM and SaM therefore SeP. Related to celarent . (Contributed by David A. Wheeler, 27-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	cesare.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
	cesare.min	⊢ ∀ 𝑥 ( 𝜒 → 𝜓 )
Assertion	cesare	⊢ ∀ 𝑥 ( 𝜒 → ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		cesare.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
2		cesare.min	⊢ ∀ 𝑥 ( 𝜒 → 𝜓 )
3		con2	⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ¬ 𝜑 ) )
4	3	alimi	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ∀ 𝑥 ( 𝜓 → ¬ 𝜑 ) )
5	1 4	ax-mp	⊢ ∀ 𝑥 ( 𝜓 → ¬ 𝜑 )
6	5 2	celarent	⊢ ∀ 𝑥 ( 𝜒 → ¬ 𝜑 )