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Mirrors > Home > MPE Home > Th. List > fesapo | Structured version Visualization version GIF version |
Description: "Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
fesapo.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
fesapo.min | ⊢ ∀𝑥(𝜓 → 𝜒) |
fesapo.e | ⊢ ∃𝑥𝜓 |
Ref | Expression |
---|---|
fesapo | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fesapo.e | . 2 ⊢ ∃𝑥𝜓 | |
2 | fesapo.min | . . . 4 ⊢ ∀𝑥(𝜓 → 𝜒) | |
3 | 2 | spi 2042 | . . 3 ⊢ (𝜓 → 𝜒) |
4 | fesapo.maj | . . . . 5 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
5 | 4 | spi 2042 | . . . 4 ⊢ (𝜑 → ¬ 𝜓) |
6 | 5 | con2i 133 | . . 3 ⊢ (𝜓 → ¬ 𝜑) |
7 | 3, 6 | jca 553 | . 2 ⊢ (𝜓 → (𝜒 ∧ ¬ 𝜑)) |
8 | 1, 7 | eximii 1754 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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