Theorem con2i 630

Description: A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)

Hypothesis

Ref	Expression
con2i.a	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
con2i	⊢ (𝜓 → ¬ 𝜑)

Proof of Theorem con2i

Step	Hyp	Ref	Expression
1		con2i.a	. 2 ⊢ (𝜑 → ¬ 𝜓)
2		id 19	. 2 ⊢ (𝜓 → 𝜓)
3	1, 2	nsyl3 629	1 ⊢ (𝜓 → ¬ 𝜑)

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:  nsyl  631  notnot  632  imanim  692  imnan  694  pm4.53r  756  ioran  757  pm3.1  759  oranim  786  xornbi  1428  exalim  1548  exnalim  1692  festino  2184  calemes  2194  fresison  2196  calemos  2197  fesapo  2198  nner  2404  necon2ai  2454  necon2bi  2455  neneqad  2479  ralexim  2522  rexalim  2523  eueq3dc  2977  elndif  3328  ssddif  3438  unssdif  3439  n0i  3497  preleq  4647  dcextest  4673  dmsn0el  5198  funtpg  5372  ftpg  5827  acexmidlemab  6001  reldmtpos  6405  nntri2  6648  nntri3  6651  nndceq  6653  inffiexmid  7076  ctssdccl  7286  mkvprop  7333  elni2  7509  renfdisj  8214  sup3exmid  9112  fzdisj  10256  sumrbdclem  11896  prodrbdclem  12090  lgsval2lem  15697  g0wlk0  16091