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Metamath Proof Explorer

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 )