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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 2553 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 2042 | . . . 4 ⊢ (𝜓 → 𝜒) |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
5 | simpl 472 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
6 | 4, 5 | jca 553 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜑)) |
7 | 1, 6 | eximii 1754 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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