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Mirrors > Home > ILE Home > Th. List > datisi | GIF version |
Description: "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
Ref | Expression |
---|---|
datisi.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
datisi.min | ⊢ ∃𝑥(𝜑 ∧ 𝜒) |
Ref | Expression |
---|---|
datisi | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | datisi.min | . 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒) | |
2 | simpr 109 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜒) | |
3 | datisi.maj | . . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓) | |
4 | 3 | spi 1516 | . . . 4 ⊢ (𝜑 → 𝜓) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜓) |
6 | 2, 5 | jca 304 | . 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓)) |
7 | 1, 6 | eximii 1582 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1333 ∃wex 1472 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ferison 2118 |
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