Theorem barbara 2301

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 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	|- A.x(ph -> ps)
barbara.min	|- A.x(ch -> ph)

Assertion

Ref	Expression
barbara	|- A.x(ch -> ps)

Proof of Theorem barbara

Step	Hyp	Ref	Expression
1		barbara.min	. 2 |- A.x(ch -> ph)
2		barbara.maj	. 2 |- A.x(ph -> ps)
3		alsyl 1615	. 2 |- ((A.x(ch -> ph) /\ A.x(ph -> ps)) -> A.x(ch -> ps))
4	1, 2, 3	mp2an 653	1 |- A.x(ch -> ps)

Syntax hints:   -> wi 4  A.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