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Mirrors > Home > MPE Home > Th. List > felapton | Structured version Visualization version GIF version |
Description: "Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
felapton.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
felapton.min | ⊢ ∀𝑥(𝜑 → 𝜒) |
felapton.e | ⊢ ∃𝑥𝜑 |
Ref | Expression |
---|---|
felapton | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | felapton.e | . 2 ⊢ ∃𝑥𝜑 | |
2 | felapton.min | . . . 4 ⊢ ∀𝑥(𝜑 → 𝜒) | |
3 | 2 | spi 2054 | . . 3 ⊢ (𝜑 → 𝜒) |
4 | felapton.maj | . . . 4 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
5 | 4 | spi 2054 | . . 3 ⊢ (𝜑 → ¬ 𝜓) |
6 | 3, 5 | jca 554 | . 2 ⊢ (𝜑 → (𝜒 ∧ ¬ 𝜓)) |
7 | 1, 6 | eximii 1764 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∀wal 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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