Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > baroco | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
baroco.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
baroco.min | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Ref | Expression |
---|---|
baroco | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baroco.maj | . . 3 ⊢ ∀𝑥(𝜑 → 𝜓) | |
2 | con3 156 | . . . . 5 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | |
3 | 2 | anim2d 615 | . . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑))) |
4 | 3 | alimi 1818 | . . 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 1840 | . 2 ⊢ (∀𝑥((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒 ∧ ¬ 𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑))) | |
8 | 5, 6, 7 | mp2 9 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |