Theorem datisi

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps , and some ph is ch , therefore some ch is ps . In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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