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Theorem barbara 2301
Description: "Barbara", one of the fundamental syllogisms of Aristotelian logic. All φ is ψ, and all χ is φ, therefore all χ is ψ. (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 x(x Hx M) (all men are mortal) and x(x = Sx H) (Socrates is a man) therefore x(x = Sx M) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 15. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1615. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1615, http://plato.stanford.edu/entries/aristotle-logic/ 1615, and https://en.wikipedia.org/wiki/Syllogism 1615. (Contributed by David A. Wheeler, 24-Aug-2016.)
Hypotheses
Ref Expression
barbara.maj x(φψ)
barbara.min x(χφ)
Assertion
Ref Expression
barbara x(χψ)

Proof of Theorem barbara
StepHypRef Expression
1 barbara.min . 2 x(χφ)
2 barbara.maj . 2 x(φψ)
3 alsyl 1615 . 2 ((x(χφ) x(φψ)) → x(χψ))
41, 2, 3mp2an 653 1 x(χψ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557
This theorem depends on definitions:  df-bi 177  df-an 360
This theorem is referenced by:  celarent  2302  barbari  2305
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