Theorem cesare 2569

Description: "Cesare", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ps, therefore no ch is ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2564. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)

Hypotheses

Ref	Expression
cesare.maj	|- A. x ( ph -> -. ps )
cesare.min	|- A. x ( ch -> ps )

Assertion

Ref	Expression
cesare	|- A. x ( ch -> -. ph )

Proof of Theorem cesare

Step	Hyp	Ref	Expression
1		cesare.maj	. . . 4 |- A. x ( ph -> -. ps )
2	1	spi 2054	. . 3 |- ( ph -> -. ps )
3		cesare.min	. . . 4 |- A. x ( ch -> ps )
4	3	spi 2054	. . 3 |- ( ch -> ps )
5	2, 4	nsyl3 133	. 2 |- ( ch -> -. ph )
6	5	ax-gen 1722	1 |- A. x ( ch -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

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-ex 1705

This theorem is referenced by: (None)