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Mirrors > Home > MPE Home > Th. List > darapti | Structured version Visualization version GIF version |
Description: "Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) |
Ref | Expression |
---|---|
darapti.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
darapti.min | ⊢ ∀𝑥(𝜑 → 𝜒) |
darapti.e | ⊢ ∃𝑥𝜑 |
Ref | Expression |
---|---|
darapti | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | darapti.e | . 2 ⊢ ∃𝑥𝜑 | |
2 | darapti.min | . . . 4 ⊢ ∀𝑥(𝜑 → 𝜒) | |
3 | 2 | spi 2208 | . . 3 ⊢ (𝜑 → 𝜒) |
4 | darapti.maj | . . . 4 ⊢ ∀𝑥(𝜑 → 𝜓) | |
5 | 4 | spi 2208 | . . 3 ⊢ (𝜑 → 𝜓) |
6 | 3, 5 | jca 501 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜓)) |
7 | 1, 6 | eximii 1912 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1629 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1853 |
This theorem is referenced by: (None) |
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