Theorem bamalip 2324

Description: "Bamalip", one of the syllogisms of Aristotelian logic. All ph is ps, all ps is ch, and ph exist, therefore some ch is ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2305. (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
bamalip	|- E.x(ch /\ ph)

Proof of Theorem bamalip

Step	Hyp	Ref	Expression
1		bamalip.e	. 2 |- E.xph
2		bamalip.maj	. . . . . 6 |- A.x(ph -> ps)
3	2	spi 1753	. . . . 5 |- (ph -> ps)
4		bamalip.min	. . . . . 6 |- A.x(ps -> ch)
5	4	spi 1753	. . . . 5 |- (ps -> ch)
6	3, 5	syl 15	. . . 4 |- (ph -> ch)
7	6	ancri 535	. . 3 |- (ph -> (ch /\ ph))
8	7	eximi 1576	. 2 |- (E.xph -> E.x(ch /\ ph))
9	1, 8	ax-mp 5	1 |- E.x(ch /\ ph)

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)