Theorem pm2.65i 629

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 616	. 2 |- ( ph -> -. ph )
4		pm2.01 606	. 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 604  ax-in2 605

This theorem is referenced by:  mt2  630  mto  652  pm5.19  696  noel  3413  0nelop  4226  elirr  4518  en2lp  4531  soirri  4998  canth  5796  0neqopab  5887  fzp1disj  10015  fzonel  10095  fzouzdisj  10115  lgsval2lem  13551  bj-imnimnn  13619  nnnotnotr  13872