Theorem felapton 2317

Description: "Felapton", one of the syllogisms of Aristotelian logic. No ph is ps, all ph is ch, and some ph exist, therefore some ch is not ps. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
felapton.maj	|- A.x(ph -> -. ps)
felapton.min	|- A.x(ph -> ch)
felapton.e	|- E.xph

Assertion

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

Proof of Theorem felapton

Step	Hyp	Ref	Expression
1		felapton.e	. 2 |- E.xph
2		felapton.min	. . . . 5 |- A.x(ph -> ch)
3	2	spi 1753	. . . 4 |- (ph -> ch)
4		felapton.maj	. . . . 5 |- A.x(ph -> -. ps)
5	4	spi 1753	. . . 4 |- (ph -> -. ps)
6	3, 5	jca 518	. . 3 |- (ph -> (ch /\ -. ps))
7	6	eximi 1576	. 2 |- (E.xph -> E.x(ch /\ -. ps))
8	1, 7	ax-mp 5	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)