Theorem cesaro 2122

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. x ch

Assertion

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

Proof of Theorem cesaro

Step	Hyp	Ref	Expression
1		cesaro.e	. 2 |- E. x ch
2		cesaro.maj	. . . . 5 |- A. x ( ph -> -. ps )
3	2	spi 1524	. . . 4 |- ( ph -> -. ps )
4		cesaro.min	. . . . 5 |- A. x ( ch -> ps )
5	4	spi 1524	. . . 4 |- ( ch -> ps )
6	3, 5	nsyl3 616	. . 3 |- ( ch -> -. ph )
7	6	ancli 321	. 2 |- ( ch -> ( ch /\ -. ph ) )
8	1, 7	eximii 1590	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)