Theorem fesapo 2165

Description: "Fesapo", one of the syllogisms of Aristotelian logic. No ph is ps, all ps is ch, and ps exist, therefore some ch is not ph. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem fesapo

Step	Hyp	Ref	Expression
1		fesapo.e	. 2 |- E. x ps
2		fesapo.min	. . . 4 |- A. x ( ps -> ch )
3	2	spi 1550	. . 3 |- ( ps -> ch )
4		fesapo.maj	. . . . 5 |- A. x ( ph -> -. ps )
5	4	spi 1550	. . . 4 |- ( ph -> -. ps )
6	5	con2i 628	. . 3 |- ( ps -> -. ph )
7	3, 6	jca 306	. 2 |- ( ps -> ( ch /\ -. ph ) )
8	1, 7	eximii 1616	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1362   E.wex 1506

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)