Theorem pm2.65i 642

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 629	. 2 |- ( ph -> -. ph )
4		pm2.01 619	. 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 617  ax-in2 618

This theorem is referenced by:  mt2  643  mto  666  pm5.19  711  noel  3495  0nelop  4333  elirr  4632  en2lp  4645  soirri  5122  canth  5951  0neqopab  6048  fzp1disj  10272  fzonel  10353  fzouzdisj  10374  4sqlem17  12925  lgsval2lem  15683  bj-imnimnn  16060  nnnotnotr  16311