Theorem baroco 2676

Description: "Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. 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.)

Hypotheses

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

Assertion

Ref	Expression
baroco	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem baroco

Step	Hyp	Ref	Expression
1		baroco.maj	. . 3 ⊢ ∀𝑥(𝜑 → 𝜓)
2		con3 153	. . . . 5 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
3	2	anim2d 612	. . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
4	3	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
5	1, 4	ax-mp 5	. 2 ⊢ ∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑))
6		baroco.min	. 2 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
7		exim 1834	. 2 ⊢ (∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒 ∧ ¬ 𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑)))
8	5, 6, 7	mp2 9	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by: (None)