Theorem camestros 2574

Description: "Camestros", one of the syllogisms of Aristotelian logic. All ph is ps, no ch is ps, and ch exist, therefore some ch is not ph. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
camestros.maj	|- A. x ( ph -> ps )
camestros.min	|- A. x ( ch -> -. ps )
camestros.e	|- E. x ch

Assertion

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

Proof of Theorem camestros

Step	Hyp	Ref	Expression
1		camestros.e	. 2 |- E. x ch
2		camestros.min	. . . . 5 |- A. x ( ch -> -. ps )
3	2	spi 2054	. . . 4 |- ( ch -> -. ps )
4		camestros.maj	. . . . 5 |- A. x ( ph -> ps )
5	4	spi 2054	. . . 4 |- ( ph -> ps )
6	3, 5	nsyl 135	. . 3 |- ( ch -> -. ph )
7	6	ancli 574	. 2 |- ( ch -> ( ch /\ -. ph ) )
8	1, 7	eximii 1764	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)