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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	\|- E. x ( ph /\ -. ps )
	bocardo.min	\|- A. x ( ph -> ch )
Assertion	bocardo	\|- E. x ( ch /\ -. ps )

Proof

Step	Hyp	Ref	Expression
1		bocardo.maj	\|- E. x ( ph /\ -. ps )
2		bocardo.min	\|- A. x ( ph -> ch )
3	1 2	disamis	\|- E. x ( ch /\ -. ps )