Theorem cesaro 2311

Description: "Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ps, and ch exist, therefore some ch is not ph. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem cesaro

Step	Hyp	Ref	Expression
1		cesaro.e	. 2 |- E.xch
2		cesaro.maj	. . . . . 6 |- A.x(ph -> -. ps)
3	2	spi 1753	. . . . 5 |- (ph -> -. ps)
4		cesaro.min	. . . . . 6 |- A.x(ch -> ps)
5	4	spi 1753	. . . . 5 |- (ch -> ps)
6	3, 5	nsyl3 111	. . . 4 |- (ch -> -. ph)
7	6	ancli 534	. . 3 |- (ch -> (ch /\ -. ph))
8	7	eximi 1576	. 2 |- (E.xch -> E.x(ch /\ -. ph))
9	1, 8	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)