Theorem pm2.65i 644

Description: Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)

Hypotheses

Ref	Expression
pm2.65i.1	|- ( ph -> ps )
pm2.65i.2	|- ( ph -> -. ps )

Assertion

Ref	Expression
pm2.65i	|- -. ph

Proof of Theorem pm2.65i

Step	Hyp	Ref	Expression
1		pm2.65i.2	. . 3 |- ( ph -> -. ps )
2		pm2.65i.1	. . 3 |- ( ph -> ps )
3	1, 2	nsyl3 631	. 2 |- ( ph -> -. ph )
4		pm2.01 621	. 2 |- ( ( ph -> -. ph ) -> -. ph )
5	3, 4	ax-mp 5	1 |- -. ph

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 619  ax-in2 620

This theorem is referenced by:  mt2  645  mto  668  pm5.19  714  noel  3512  0nelop  4364  elirr  4663  en2lp  4676  soirri  5157  canth  6001  0neqopab  6098  fczsupp0  6459  fzp1disj  10414  fzonel  10495  fzouzdisj  10516  hashfibclem  11206  4sqlem17  13105  lgsval2lem  15883  bj-imnimnn  16510  nnnotnotr  16760