Theorem fresison 2059

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 108	. . 3 |- ( ( ps /\ ch ) -> ch )
3		fresison.maj	. . . . . 6 |- A. x ( ph -> -. ps )
4	3	spi 1469	. . . . 5 |- ( ph -> -. ps )
5	4	con2i 589	. . . 4 |- ( ps -> -. ph )
6	5	adantr 270	. . 3 |- ( ( ps /\ ch ) -> -. ph )
7	2, 6	jca 300	. 2 |- ( ( ps /\ ch ) -> ( ch /\ -. ph ) )
8	1, 7	eximii 1533	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)