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Theorem barbara 2301
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 (all men are mortal) and (Socrates is a man) therefore (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,, and (Contributed by David A. Wheeler, 24-Aug-2016.)
Ref Expression
Ref Expression

Proof of Theorem barbara
StepHypRef Expression
1 barbara.min . 2
2 barbara.maj . 2
3 alsyl 1615 . 2
41, 2, 3mp2an 653 1
Colors of variables: wff setvar class
Syntax hints:   wi 4  wal 1540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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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