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  3500  0nelop  4346  elirr  4645  en2lp  4658  soirri  5138  canth  5979  0neqopab  6076  fczsupp0  6437  fzp1disj  10360  fzonel  10441  fzouzdisj  10462  4sqlem17  13043  lgsval2lem  15812  bj-imnimnn  16439  nnnotnotr  16689