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 )