Theorem cesaro 2153

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 1550	. . . 4 |- ( ph -> -. ps )
4		cesaro.min	. . . . 5 |- A. x ( ch -> ps )
5	4	spi 1550	. . . 4 |- ( ch -> ps )
6	3, 5	nsyl3 627	. . 3 |- ( ch -> -. ph )
7	6	ancli 323	. 2 |- ( ch -> ( ch /\ -. ph ) )
8	1, 7	eximii 1616	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)