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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 2075 | . . . 4 ⊢ ∀𝑥(𝜒 → 𝜓) |
5 | 4 | spi 1501 | . . 3 ⊢ (𝜒 → 𝜓) |
6 | 5 | ancli 321 | . 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓)) |
7 | 1, 6 | eximii 1566 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1314 ∃wex 1453 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: celaront 2080 |
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