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

Theorem baroco 2249
Description: "Baroco", one of the syllogisms of Aristotelian logic. All  ph is  ps, and some  ch is not  ps, therefore some  ch is not  ph. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
baroco.maj  |-  A. x
( ph  ->  ps )
baroco.min  |-  E. x
( ch  /\  -.  ps )
Assertion
Ref Expression
baroco  |-  E. x
( ch  /\  -.  ph )

Proof of Theorem baroco
StepHypRef Expression
1 baroco.min . 2  |-  E. x
( ch  /\  -.  ps )
2 baroco.maj . . . . . 6  |-  A. x
( ph  ->  ps )
32spi 1738 . . . . 5  |-  ( ph  ->  ps )
43con3i 127 . . . 4  |-  ( -. 
ps  ->  -.  ph )
54anim2i 552 . . 3  |-  ( ( ch  /\  -.  ps )  ->  ( ch  /\  -.  ph ) )
65eximi 1563 . 2  |-  ( E. x ( ch  /\  -.  ps )  ->  E. x
( ch  /\  -.  ph ) )
71, 6ax-mp 8 1  |-  E. x
( ch  /\  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
  Copyright terms: Public domain W3C validator