Theorem festino 2571

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	. . . . 5 |- A. x ( ph -> -. ps )
3	2	spi 2054	. . . 4 |- ( ph -> -. ps )
4	3	con2i 134	. . 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)