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Theorem bocardo 2268
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 2266; 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
StepHypRef Expression
1 bocardo.maj . 2  |-  E. x
( ph  /\  -.  ps )
2 bocardo.min . 2  |-  A. x
( ph  ->  ch )
31, 2disamis 2266 1  |-  E. x
( ch  /\  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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