Theorem barbara 2112

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 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1623. There are a legion of sources for Barbara, including https://www.friesian.com/aristotl.htm 1623, https://plato.stanford.edu/entries/aristotle-logic/ 1623, and https://en.wikipedia.org/wiki/Syllogism 1623. (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 1623	. 2 |- ( ( A. x ( ch -> ph ) /\ A. x ( ph -> ps ) ) -> A. x ( ch -> ps ) )
4	1, 2, 3	mp2an 423	1 |- A. x ( ch -> ps )

Syntax hints:   -> wi 4   A.wal 1341

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 1435  ax-gen 1437

This theorem is referenced by:  celarent  2113  barbari  2116