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Mirrors > Home > ILE Home > Th. List > calemos | GIF version |
Description: "Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
calemos.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
calemos.min | ⊢ ∀𝑥(𝜓 → ¬ 𝜒) |
calemos.e | ⊢ ∃𝑥𝜒 |
Ref | Expression |
---|---|
calemos | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | calemos.e | . 2 ⊢ ∃𝑥𝜒 | |
2 | calemos.min | . . . . . 6 ⊢ ∀𝑥(𝜓 → ¬ 𝜒) | |
3 | 2 | spi 1469 | . . . . 5 ⊢ (𝜓 → ¬ 𝜒) |
4 | 3 | con2i 589 | . . . 4 ⊢ (𝜒 → ¬ 𝜓) |
5 | calemos.maj | . . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓) | |
6 | 5 | spi 1469 | . . . 4 ⊢ (𝜑 → 𝜓) |
7 | 4, 6 | nsyl 590 | . . 3 ⊢ (𝜒 → ¬ 𝜑) |
8 | 7 | ancli 316 | . 2 ⊢ (𝜒 → (𝜒 ∧ ¬ 𝜑)) |
9 | 1, 8 | eximii 1533 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∀wal 1282 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: (None) |
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