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Theorem barbara 2747
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(𝑥𝐻𝑥𝑀) (all men are mortal) and 𝑥(𝑥 = 𝑆𝑥𝐻) (Socrates is a man) therefore 𝑥(𝑥 = 𝑆𝑥𝑀) (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 1997. There are a legion of sources for Barbara, including,, and (Contributed by David A. Wheeler, 24-Aug-2016.)
Ref Expression
barbara.maj 𝑥(𝜑𝜓)
barbara.min 𝑥(𝜒𝜑)
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 1997 . 2 ((∀𝑥(𝜒𝜑) ∧ ∀𝑥(𝜑𝜓)) → ∀𝑥(𝜒𝜓))
41, 2, 3mp2an 685 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910
This theorem depends on definitions:  df-bi 199  df-an 387
This theorem is referenced by:  celarent  2748  barbari  2753  barbariALT  2754
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