Theorem nsyli 106

Description: A negated syllogism inference.

Hypotheses

Ref	Expression
nsyli.1	|- (ph -> (ps -> ch))
nsyli.2	|- (th -> -. ch)

Assertion

Ref	Expression
nsyli	|- (ph -> (th -> -. ps))

Proof of Theorem nsyli

Step	Hyp	Ref	Expression
1		nsyli.1	. . 3 |- (ph -> (ps -> ch))
2	1	con3d 87	. 2 |- (ph -> (-. ch -> -. ps))
3		nsyli.2	. 2 |- (th -> -. ch)
4	2, 3	syl5 22	1 |- (ph -> (th -> -. ps))

Syntax hints:  -. wn 1   -> wi 2

This theorem is referenced by:  tz7.7 2224  tz7.48-2 2995  nnmord 3189  php 3409  nndomo 3416  isfinite2 3437  eirrv 3449  setind 3492  zornlem3 3605  infxpidmlem10 4942

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

metamath.org