Theorem baroco 2572

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	. . . . 5 |- A. x ( ph -> ps )
3	2	spi 2054	. . . 4 |- ( ph -> ps )
4	3	con3i 150	. . 3 |- ( -. ps -> -. ph )
5	4	anim2i 593	. 2 |- ( ( ch /\ -. ps ) -> ( ch /\ -. ph ) )
6	1, 5	eximii 1764	1 |- E. x ( ch /\ -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)