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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 2565 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Ref | Expression |
---|---|
dimatis.maj | ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
dimatis.min | ⊢ ∀𝑥(𝜓 → 𝜒) |
Ref | Expression |
---|---|
dimatis | ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dimatis.maj | . 2 ⊢ ∃𝑥(𝜑 ∧ 𝜓) | |
2 | dimatis.min | . . . . 5 ⊢ ∀𝑥(𝜓 → 𝜒) | |
3 | 2 | spi 2054 | . . . 4 ⊢ (𝜓 → 𝜒) |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
5 | simpl 473 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
6 | 4, 5 | jca 554 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜑)) |
7 | 1, 6 | eximii 1764 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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