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Theorem barbara 2073
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 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1597. There are a legion of sources for Barbara, including 1597, 1597, and 1597. (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 1597 . 2 ((∀𝑥(𝜒𝜑) ∧ ∀𝑥(𝜑𝜓)) → ∀𝑥(𝜒𝜓))
41, 2, 3mp2an 420 1 𝑥(𝜒𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408
This theorem is referenced by:  celarent  2074  barbari  2077
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