Theorem calemos 2060

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 1469	. . . . 5 |- ( ps -> -. ch )
4	3	con2i 589	. . . 4 |- ( ch -> -. ps )
5		calemos.maj	. . . . 5 |- A. x ( ph -> ps )
6	5	spi 1469	. . . 4 |- ( ph -> ps )
7	4, 6	nsyl 590	. . 3 |- ( ch -> -. ph )
8	7	ancli 316	. 2 |- ( ch -> ( ch /\ -. ph ) )
9	1, 8	eximii 1533	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)