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Theorem barbara 1995
Description: "Barbara", one of the fundamental syllogisms of Aristotelian logic. All is , and all is , therefore all is . (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  H  M (all men are mortal) and  S  H (Socrates is a man) therefore  S  M (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1523. 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
barbara.min
Assertion
Ref Expression
barbara

Proof of Theorem barbara
StepHypRef Expression
1 barbara.min . 2
2 barbara.maj . 2
3 alsyl 1523 . 2
41, 2, 3mp2an 402 1
Colors of variables: wff set class
Syntax hints:   wi 4  wal 1240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335
This theorem is referenced by:  celarent  1996  barbari  1999
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