Theorem pm2.65i 644

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 631	. 2 |- ( ph -> -. ph )
4		pm2.01 621	. 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 619  ax-in2 620

This theorem is referenced by:  mt2  645  mto  668  pm5.19  714  noel  3511  0nelop  4363  elirr  4662  en2lp  4675  soirri  5156  canth  6000  0neqopab  6097  fczsupp0  6458  fzp1disj  10413  fzonel  10494  fzouzdisj  10515  hashfibclem  11202  4sqlem17  13101  lgsval2lem  15875  bj-imnimnn  16502  nnnotnotr  16752