Theorem felapton 2128

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. x ph

Assertion

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

Proof of Theorem felapton

Step	Hyp	Ref	Expression
1		felapton.e	. 2 |- E. x ph
2		felapton.min	. . . 4 |- A. x ( ph -> ch )
3	2	spi 1524	. . 3 |- ( ph -> ch )
4		felapton.maj	. . . 4 |- A. x ( ph -> -. ps )
5	4	spi 1524	. . 3 |- ( ph -> -. ps )
6	3, 5	jca 304	. 2 |- ( ph -> ( ch /\ -. ps ) )
7	1, 6	eximii 1590	1 |- E. x ( ch /\ -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480

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  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)