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Mirrors > Home > ILE Home > Th. List > bocardo | GIF version |
Description: "Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2052; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bocardo.maj | ⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓) |
bocardo.min | ⊢ ∀𝑥(𝜑 → 𝜒) |
Ref | Expression |
---|---|
bocardo | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bocardo.maj | . 2 ⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓) | |
2 | bocardo.min | . 2 ⊢ ∀𝑥(𝜑 → 𝜒) | |
3 | 1, 2 | disamis 2052 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) |
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-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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