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Theorem barbara 2210
Description: "Barbara", one of the fundamental syllogisms of Aristotelian logic. All  ph is  ps, and all  ch is  ph, therefore all  ch is  ps. (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as  A. x ( x  e.  H  ->  x  e.  M ) (all men are mortal) and  A. x ( x  =  S  ->  x  e.  H ) (Socrates is a man) therefore  A. x ( x  =  S  ->  x  e.  M ) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1617. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
barbara.maj  |-  A. x
( ph  ->  ps )
barbara.min  |-  A. x
( ch  ->  ph )
Assertion
Ref Expression
barbara  |-  A. x
( ch  ->  ps )

Proof of Theorem barbara
StepHypRef Expression
1 barbara.min . 2  |-  A. x
( ch  ->  ph )
2 barbara.maj . 2  |-  A. x
( ph  ->  ps )
3 alsyl 1617 . 2  |-  ( ( A. x ( ch 
->  ph )  /\  A. x ( ph  ->  ps ) )  ->  A. x
( ch  ->  ps ) )
41, 2, 3mp2an 656 1  |-  A. x
( ch  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  celarent  2211  barbari  2214
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179  df-an 362
  Copyright terms: Public domain W3C validator