Theorem festino 2309

Description: "Festino", one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ps, therefore some ch is not ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem festino

Step	Hyp	Ref	Expression
1		festino.min	. 2 |- E.x(ch /\ ps)
2		festino.maj	. . . . . 6 |- A.x(ph -> -. ps)
3	2	spi 1753	. . . . 5 |- (ph -> -. ps)
4	3	con2i 112	. . . 4 |- (ps -> -. ph)
5	4	anim2i 552	. . 3 |- ((ch /\ ps) -> (ch /\ -. ph))
6	5	eximi 1576	. 2 |- (E.x(ch /\ ps) -> E.x(ch /\ -. ph))
7	1, 6	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)