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Metamath Proof Explorer

Theorem fesapo

Description: "Fesapo", one of the syllogisms of Aristotelian logic. No ph is ps , all ps is ch , and ps exist, therefore some ch is not ph . In Aristotelian notation, EAO-4: PeM and MaS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		fesapo.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
2		fesapo.min	⊢ ∀ 𝑥 ( 𝜓 → 𝜒 )
3		fesapo.e	⊢ ∃ 𝑥 𝜓
4		con2	⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ¬ 𝜑 ) )
5	4	alimi	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ∀ 𝑥 ( 𝜓 → ¬ 𝜑 ) )
6	1 5	ax-mp	⊢ ∀ 𝑥 ( 𝜓 → ¬ 𝜑 )
7	6 2 3	felapton	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )