Theorem darapti 2580

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. x ph

Assertion

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

Proof of Theorem darapti

Step	Hyp	Ref	Expression
1		darapti.e	. 2 |- E. x ph
2		darapti.min	. . . 4 |- A. x ( ph -> ch )
3	2	spi 2054	. . 3 |- ( ph -> ch )
4		darapti.maj	. . . 4 |- A. x ( ph -> ps )
5	4	spi 2054	. . 3 |- ( ph -> ps )
6	3, 5	jca 554	. 2 |- ( ph -> ( ch /\ ps ) )
7	1, 6	eximii 1764	1 |- E. x ( ch /\ ps )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)