Theorem fresison

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

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

Proof

Step	Hyp	Ref	Expression
1		fresison.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
2		fresison.min	⊢ ∃ 𝑥 ( 𝜓 ∧ 𝜒 )
3		exancom	⊢ ( ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜒 ∧ 𝜓 ) )
4	2 3	mpbi	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )
5	1 4	festino	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )