Theorem baroco 2310

Description: "Baroco", one of the syllogisms of Aristotelian logic. All ph is ps, and some ch is not ps, therefore some ch is not ph. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem baroco

Step	Hyp	Ref	Expression
1		baroco.min	. 2 |- E.x(ch /\ -. ps)
2		baroco.maj	. . . . . 6 |- A.x(ph -> ps)
3	2	spi 1753	. . . . 5 |- (ph -> ps)
4	3	con3i 127	. . . 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)