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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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
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 | . . 3 ⊢ ∀𝑥(𝜑 → 𝜓) | |
3 | barbari.min | . . 3 ⊢ ∀𝑥(𝜒 → 𝜑) | |
4 | 2, 3 | barbara 2659 | . 2 ⊢ ∀𝑥(𝜒 → 𝜓) |
5 | 1, 4 | barbarilem 2664 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: celaront 2667 bamalip 2688 |
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