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Metamath Proof Explorer

Theorem bocardo

Description: "Bocardo", one of the syllogisms of Aristotelian logic. Some ph is not ps , and all ph is ch , therefore some ch is not ps . Instance of disamis . 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". (Contributed by David A. Wheeler, 28-Aug-2016)

	Ref	Expression
Hypotheses	bocardo.maj	⊢ ∃ 𝑥 ( 𝜑 ∧ ¬ 𝜓 )
	bocardo.min	⊢ ∀ 𝑥 ( 𝜑 → 𝜒 )
Assertion	bocardo	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜓 )

Proof

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