Theorem fresison 2137

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

Hypotheses

Ref	Expression
fresison.maj	|- A. x ( ph -> -. ps )
fresison.min	|- E. x ( ps /\ ch )

Assertion

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

Proof of Theorem fresison

Step	Hyp	Ref	Expression
1		fresison.min	. 2 |- E. x ( ps /\ ch )
2		simpr 109	. . 3 |- ( ( ps /\ ch ) -> ch )
3		fresison.maj	. . . . . 6 |- A. x ( ph -> -. ps )
4	3	spi 1529	. . . . 5 |- ( ph -> -. ps )
5	4	con2i 622	. . . 4 |- ( ps -> -. ph )
6	5	adantr 274	. . 3 |- ( ( ps /\ ch ) -> -. ph )
7	2, 6	jca 304	. 2 |- ( ( ps /\ ch ) -> ( ch /\ -. ph ) )
8	1, 7	eximii 1595	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)