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Mirrors > Home > ILE Home > Th. List > barbari | GIF version |
Description: "Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.) |
Ref | Expression |
---|---|
barbari.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
barbari.min | ⊢ ∀𝑥(𝜒 → 𝜑) |
barbari.e | ⊢ ∃𝑥𝜒 |
Ref | Expression |
---|---|
barbari | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | barbari.e | . 2 ⊢ ∃𝑥𝜒 | |
2 | barbari.maj | . . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓) | |
3 | barbari.min | . . . . 5 ⊢ ∀𝑥(𝜒 → 𝜑) | |
4 | 2, 3 | barbara 2097 | . . . 4 ⊢ ∀𝑥(𝜒 → 𝜓) |
5 | 4 | spi 1516 | . . 3 ⊢ (𝜒 → 𝜓) |
6 | 5 | ancli 321 | . 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓)) |
7 | 1, 6 | eximii 1581 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
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: celaront 2102 |
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