Theorem datisi 2313

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps, and some ph is ch, therefore some ch is ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem datisi

Step	Hyp	Ref	Expression
1		datisi.min	. 2 |- E.x(ph /\ ch)
2		simpr 447	. . . 4 |- ((ph /\ ch) -> ch)
3		datisi.maj	. . . . . 6 |- A.x(ph -> ps)
4	3	spi 1753	. . . . 5 |- (ph -> ps)
5	4	adantr 451	. . . 4 |- ((ph /\ ch) -> ps)
6	2, 5	jca 518	. . 3 |- ((ph /\ ch) -> (ch /\ ps))
7	6	eximi 1576	. 2 |- (E.x(ph /\ ch) -> 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:  ferison  2315