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Metamath Proof Explorer

Theorem calemes

Description: "Calemes", one of the syllogisms of Aristotelian logic. All ph is ps , and no ps is ch , therefore no ch is ph . In Aristotelian notation, AEE-4: PaM and MeS therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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