Theorem calemos 2199

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 1585	. . . . 5 |- ( ps -> -. ch )
4	3	con2i 632	. . . 4 |- ( ch -> -. ps )
5		calemos.maj	. . . . 5 |- A. x ( ph -> ps )
6	5	spi 1585	. . . 4 |- ( ph -> ps )
7	4, 6	nsyl 633	. . 3 |- ( ch -> -. ph )
8	7	ancli 323	. 2 |- ( ch -> ( ch /\ -. ph ) )
9	1, 8	eximii 1651	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1396   E.wex 1541

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)