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Mirrors > Home > MPE Home > Th. List > cesaro | Structured version Visualization version GIF version |
Description: "Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
cesaro.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
cesaro.min | ⊢ ∀𝑥(𝜒 → 𝜓) |
cesaro.e | ⊢ ∃𝑥𝜒 |
Ref | Expression |
---|---|
cesaro | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cesaro.e | . 2 ⊢ ∃𝑥𝜒 | |
2 | cesaro.maj | . . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
3 | 2 | spi 2054 | . . . 4 ⊢ (𝜑 → ¬ 𝜓) |
4 | cesaro.min | . . . . 5 ⊢ ∀𝑥(𝜒 → 𝜓) | |
5 | 4 | spi 2054 | . . . 4 ⊢ (𝜒 → 𝜓) |
6 | 3, 5 | nsyl3 133 | . . 3 ⊢ (𝜒 → ¬ 𝜑) |
7 | 6 | ancli 574 | . 2 ⊢ (𝜒 → (𝜒 ∧ ¬ 𝜑)) |
8 | 1, 7 | eximii 1764 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
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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