Theorem pm2.65i 601

Description: Inference rule 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 589	. 2 |- ( ph -> -. ph )
4		pm2.01 579	. 2 |- ( ( ph -> -. ph ) -> -. ph )
5	3, 4	ax-mp 7	1 |- -. ph

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 577  ax-in2 578

This theorem is referenced by:  mt2  602  mto  621  pm5.19  655  noel  3279  0nelop  4049  elirr  4330  en2lp  4343  soirri  4793  0neqopab  5651  fzp1disj  9424  fzonel  9499  fzouzdisj  9519