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  4334  elirr  4633  en2lp  4646  soirri  5123  canth  5958  0neqopab  6055  fzp1disj  10288  fzonel  10369  fzouzdisj  10390  4sqlem17  12945  lgsval2lem  15704  bj-imnimnn  16161  nnnotnotr  16412