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| 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 2665 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 2665 | . 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒) | 
| 4 | exancom 1861 | . 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 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) | 
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