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Metamath Proof Explorer

Theorem camestres

Description: "Camestres", one of the syllogisms of Aristotelian logic. All ph is ps , and no ch is ps , therefore no ch is ph . In Aristotelian notation, AEE-2: PaM and SeM therefore SeP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		camestres.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
2		camestres.min	⊢ ∀ 𝑥 ( 𝜒 → ¬ 𝜓 )
3		con3	⊢ ( ( 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 )