Theorem pm2.65i 648

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 635	. 2 |- ( ph -> -. ph )
4		pm2.01 625	. 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 623  ax-in2 624

This theorem is referenced by:  mt2  649  mto  672  pm5.19  718  noel  3525  0nelop  4383  elirr  4683  en2lp  4696  soirri  5177  canth  6026  0neqopab  6123  fczsupp0  6489  fzp1disj  10465  fzonel  10546  fzouzdisj  10567  hashfibclem  11260  4sqlem17  13164  lgsval2lem  16043  bj-imnimnn  16680  nnnotnotr  16930