Theorem darii

Description: "Darii", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is ph , therefore some ch is ps . In Aristotelian notation, AII-1: MaP and SiM therefore SiP. For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism . See dariiALT for a shorter proof requiring more axioms. (Contributed by David A. Wheeler, 24-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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