MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ferio Unicode version

Theorem ferio 2256
Description: "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No  ph is  ps, and some  ch is  ph, therefore some  ch is not  ps. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
ferio.maj  |-  A. x
( ph  ->  -.  ps )
ferio.min  |-  E. x
( ch  /\  ph )
Assertion
Ref Expression
ferio  |-  E. x
( ch  /\  -.  ps )

Proof of Theorem ferio
StepHypRef Expression
1 ferio.maj . 2  |-  A. x
( ph  ->  -.  ps )
2 ferio.min . 2  |-  E. x
( ch  /\  ph )
31, 2darii 2255 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
  Copyright terms: Public domain W3C validator