Theorem calemes 2581

Description: "Calemes", one of the syllogisms of Aristotelian logic. All ph is ps, and no ps is ch, therefore no ch is ph. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem calemes

Step	Hyp	Ref	Expression
1		calemes.min	. . . . 5 |- A. x ( ps -> -. ch )
2	1	spi 2054	. . . 4 |- ( ps -> -. ch )
3	2	con2i 134	. . 3 |- ( ch -> -. ps )
4		calemes.maj	. . . 4 |- A. x ( ph -> ps )
5	4	spi 2054	. . 3 |- ( ph -> ps )
6	3, 5	nsyl 135	. 2 |- ( ch -> -. ph )
7	6	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)