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Mirrors > Home > MPE Home > Th. List > dimatis | Structured version Visualization version GIF version |
Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. In Aristotelian notation, IAI-4: PiM and MaS therefore SiP. For example, "Some pets are rabbits", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2663 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
Ref | Expression |
---|---|
dimatis.maj | ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
dimatis.min | ⊢ ∀𝑥(𝜓 → 𝜒) |
Ref | Expression |
---|---|
dimatis | ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dimatis.min | . . 3 ⊢ ∀𝑥(𝜓 → 𝜒) | |
2 | dimatis.maj | . . 3 ⊢ ∃𝑥(𝜑 ∧ 𝜓) | |
3 | 1, 2 | darii 2663 | . 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒) |
4 | exancom 1859 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜑)) | |
5 | 3, 4 | mpbi 230 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: (None) |
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