Theorem darapti 2318

Description: "Darapti", one of the syllogisms of Aristotelian logic. All ph is ps, all ph is ch, and some ph exist, therefore some ch is ps. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem darapti

Step	Hyp	Ref	Expression
1		darapti.e	. 2 |- E.xph
2		darapti.min	. . . . 5 |- A.x(ph -> ch)
3	2	spi 1753	. . . 4 |- (ph -> ch)
4		darapti.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:   -> 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)