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Mirrors > Home > MPE Home > Th. List > datisi | Structured version Visualization version GIF version |
Description: "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
Ref | Expression |
---|---|
datisi.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
datisi.min | ⊢ ∃𝑥(𝜑 ∧ 𝜒) |
Ref | Expression |
---|---|
datisi | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | datisi.maj | . 2 ⊢ ∀𝑥(𝜑 → 𝜓) | |
2 | datisi.min | . . 3 ⊢ ∃𝑥(𝜑 ∧ 𝜒) | |
3 | exancom 1868 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜑)) | |
4 | 2, 3 | mpbi 229 | . 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
5 | 1, 4 | darii 2668 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀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 206 df-an 397 df-ex 1787 |
This theorem is referenced by: disamis 2684 ferison 2685 |
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