Theorem fesapo 2020

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 1429	. . 3 |- ( ps -> ch )
4		fesapo.maj	. . . . 5 |- A. x ( ph -> -. ps )
5	4	spi 1429	. . . 4 |- ( ph -> -. ps )
6	5	con2i 557	. . 3 |- ( ps -> -. ph )
7	3, 6	jca 290	. 2 |- ( ps -> ( ch /\ -. ph ) )
8	1, 7	eximii 1493	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)