Theorem bocardo 2110

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 2108; 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(𝜑 → 𝜒)

Assertion

Ref	Expression
bocardo	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem bocardo

Step	Hyp	Ref	Expression
1		bocardo.maj	. 2 ⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)
2		bocardo.min	. 2 ⊢ ∀𝑥(𝜑 → 𝜒)
3	1, 2	disamis 2108	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)