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Mirrors > Home > MPE Home > Th. List > barbari | Structured version Visualization version 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 2551 | . . . 4 ⊢ ∀𝑥(𝜒 → 𝜓) |
5 | 4 | spi 2042 | . . 3 ⊢ (𝜒 → 𝜓) |
6 | 5 | ancli 572 | . 2 ⊢ (𝜒 → (𝜒 ∧ 𝜓)) |
7 | 1, 6 | eximii 1754 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
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: celaront 2556 |
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