Theorem pm2.65i 640

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 627	. 2 |- ( ph -> -. ph )
4		pm2.01 617	. 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 615  ax-in2 616

This theorem is referenced by:  mt2  641  mto  663  pm5.19  707  noel  3455  0nelop  4282  elirr  4578  en2lp  4591  soirri  5065  canth  5878  0neqopab  5971  fzp1disj  10174  fzonel  10255  fzouzdisj  10275  4sqlem17  12603  lgsval2lem  15359  bj-imnimnn  15492  nnnotnotr  15744