Theorem pm2.65i 568

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 556	. 2 |- ( ph -> -. ph )
4		pm2.01 546	. 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 544  ax-in2 545

This theorem is referenced by:  mt2  569  mto  588  pm5.19  622  noel  3228  0nelop  3985  elirr  4266  en2lp  4278  soirri  4719  0neqopab  5550  fzp1disj  8942  fzonel  9016  fzouzdisj  9036