Theorem calemos 2145

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. x ch

Assertion

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

Proof of Theorem calemos

Step	Hyp	Ref	Expression
1		calemos.e	. 2 |- E. x ch
2		calemos.min	. . . . . 6 |- A. x ( ps -> -. ch )
3	2	spi 1536	. . . . 5 |- ( ps -> -. ch )
4	3	con2i 627	. . . 4 |- ( ch -> -. ps )
5		calemos.maj	. . . . 5 |- A. x ( ph -> ps )
6	5	spi 1536	. . . 4 |- ( ph -> ps )
7	4, 6	nsyl 628	. . 3 |- ( ch -> -. ph )
8	7	ancli 323	. 2 |- ( ch -> ( ch /\ -. ph ) )
9	1, 8	eximii 1602	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1351   E.wex 1492

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)