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Mirrors > Home > ILE Home > Th. List > dimatis | 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 2099 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 1516 | . . . 4 ⊢ (𝜓 → 𝜒) |
4 | 3 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
5 | simpl 108 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
6 | 4, 5 | jca 304 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜑)) |
7 | 1, 6 | eximii 1581 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 ∃wex 1468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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