Theorem fresison 2321

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 447	. . . 4 |- ((ps /\ ch) -> ch)
3		fresison.maj	. . . . . . 7 |- A.x(ph -> -. ps)
4	3	spi 1753	. . . . . 6 |- (ph -> -. ps)
5	4	con2i 112	. . . . 5 |- (ps -> -. ph)
6	5	adantr 451	. . . 4 |- ((ps /\ ch) -> -. ph)
7	2, 6	jca 518	. . 3 |- ((ps /\ ch) -> (ch /\ -. ph))
8	7	eximi 1576	. 2 |- (E.x(ps /\ ch) -> E.x(ch /\ -. ph))
9	1, 8	ax-mp 5	1 |- E.x(ch /\ -. ph)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)