Theorem baroco

Description: "Baroco", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is not ps , therefore some ch is not ph . In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		baroco.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
2		baroco.min	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜓 )
3		con3	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )
4	3	anim2d	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 ∧ ¬ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) )
5	4	alimi	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∀ 𝑥 ( ( 𝜒 ∧ ¬ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) )
6	1 5	ax-mp	⊢ ∀ 𝑥 ( ( 𝜒 ∧ ¬ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) )
7		exim	⊢ ( ∀ 𝑥 ( ( 𝜒 ∧ ¬ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) → ( ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜓 ) → ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 ) ) )
8	6 2 7	mp2	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )