Theorem calemos 2322

Description: "Calemos", one of the syllogisms of Aristotelian logic. All ph is ps (PaM), no ps is ch (MeS), and ch exist, therefore some ch is not ph (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem calemos

Step	Hyp	Ref	Expression
1		calemos.e	. 2 |- E.xch
2		calemos.min	. . . . . . 7 |- A.x(ps -> -. ch)
3	2	spi 1753	. . . . . 6 |- (ps -> -. ch)
4	3	con2i 112	. . . . 5 |- (ch -> -. ps)
5		calemos.maj	. . . . . 6 |- A.x(ph -> ps)
6	5	spi 1753	. . . . 5 |- (ph -> ps)
7	4, 6	nsyl 113	. . . 4 |- (ch -> -. ph)
8	7	ancli 534	. . 3 |- (ch -> (ch /\ -. ph))
9	8	eximi 1576	. 2 |- (E.xch -> E.x(ch /\ -. ph))
10	1, 9	ax-mp 5	1 |- E.x(ch /\ -. ph)

Syntax hints:  -. wn 3   -> 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)