Theorem calemos 2116

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 1516	. . . . 5 |- ( ps -> -. ch )
4	3	con2i 616	. . . 4 |- ( ch -> -. ps )
5		calemos.maj	. . . . 5 |- A. x ( ph -> ps )
6	5	spi 1516	. . . 4 |- ( ph -> ps )
7	4, 6	nsyl 617	. . 3 |- ( ch -> -. ph )
8	7	ancli 321	. 2 |- ( ch -> ( ch /\ -. ph ) )
9	1, 8	eximii 1581	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468

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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)