Theorem darii 2690

Description: "Darii", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is 𝜓. 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 2691 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.)

Hypotheses

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

Assertion

Ref	Expression
darii	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem darii

Step	Hyp	Ref	Expression
1		darii.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
2		id 23	. . . . 5 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
3	2	anim2d 624	. . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓)))
4	3	alimi 1844	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓)))
5	1, 4	ax-mp 5	. 2 ⊢ ∀𝑥((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
6		darii.min	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑)
7		exim 1867	. 2 ⊢ (∀𝑥((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓)) → (∃𝑥(𝜒 ∧ 𝜑) → ∃𝑥(𝜒 ∧ 𝜓)))
8	5, 6, 7	mp2 9	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  ferio  2692  datisi  2705  dimatis  2713