Theorem bocardo 2132

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. (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 2130; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bocardo	|- E. x ( ch /\ -. ps )

Proof of Theorem bocardo

Step	Hyp	Ref	Expression
1		bocardo.maj	. 2 |- E. x ( ph /\ -. ps )
2		bocardo.min	. 2 |- A. x ( ph -> ch )
3	1, 2	disamis 2130	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)