Theorem barbara

Description: "Barbara", one of the fundamental syllogisms of Aristotelian logic. All ph is ps , and all ch is ph , therefore all ch is ps . 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 A. x ( x e. H -> x e. M ) (all men are mortal) and A. x ( x = S -> x e. H ) (Socrates is a man) therefore A. x ( x = S -> x e. M ) (Socrates is mortal). Russell and Whitehead note that "the syllogism in Barbara is derived from [[ syl ]" (quote after Theorem *2.06 of WhiteheadRussell p. 101). Most of the proof is in alsyl . 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)

	Ref	Expression
Hypotheses	barbara.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
	barbara.min	$$⊢\forall x\left(\chi \to \phi \right)$$
Assertion	barbara	$$⊢\forall x\left(\chi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		barbara.maj	$$⊢\forall x\left(\phi \to \psi \right)$$
2		barbara.min	$$⊢\forall x\left(\chi \to \phi \right)$$
3		alsyl	$$⊢\left(\forall x\left(\chi \to \phi \right)\wedge \forall x\left(\phi \to \psi \right)\right)\to \forall x\left(\chi \to \psi \right)$$
4	2 1 3	mp2an	$$⊢\forall x\left(\chi \to \psi \right)$$