Theorem pm2.65i 634

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 621	. 2 |- ( ph -> -. ph )
4		pm2.01 611	. 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 609  ax-in2 610

This theorem is referenced by:  mt2  635  mto  657  pm5.19  701  noel  3418  0nelop  4231  elirr  4523  en2lp  4536  soirri  5003  canth  5805  0neqopab  5896  fzp1disj  10029  fzonel  10109  fzouzdisj  10129  lgsval2lem  13670  bj-imnimnn  13738  nnnotnotr  13990