Theorem bocardo 2316

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 2314; 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 2314	1 ⊢ ∃x(χ ∧ ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)