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Theorem bocardo 2058
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 2056; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bocardo.maj 𝑥(𝜑 ∧ ¬ 𝜓)
bocardo.min 𝑥(𝜑𝜒)
Assertion
Ref Expression
bocardo 𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem bocardo
StepHypRef Expression
1 bocardo.maj . 2 𝑥(𝜑 ∧ ¬ 𝜓)
2 bocardo.min . 2 𝑥(𝜑𝜒)
31, 2disamis 2056 1 𝑥(𝜒 ∧ ¬ 𝜓)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1285  wex 1424
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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