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| 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 2663 | . 2 ⊢ ∀𝑥(𝜒 → 𝜓) | 
| 5 | 1, 4 | barbarilem 2668 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 | 
| This theorem is referenced by: celaront 2671 bamalip 2692 | 
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