Theorem ferio 2304

Description: "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ph, therefore some ch is not ps. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
ferio.maj	|- A.x(ph -> -. ps)
ferio.min	|- E.x(ch /\ ph)

Assertion

Ref	Expression
ferio	|- E.x(ch /\ -. ps)

Proof of Theorem ferio

Step	Hyp	Ref	Expression
1		ferio.maj	. 2 |- A.x(ph -> -. ps)
2		ferio.min	. 2 |- E.x(ch /\ ph)
3	1, 2	darii 2303	1 |- E.x(ch /\ -. ps)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)