Theorem notbii 672

Description: Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Hypothesis

Ref	Expression
notbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
notbii	⊢ (¬ 𝜑 ↔ ¬ 𝜓)

Proof of Theorem notbii

Step	Hyp	Ref	Expression
1		notbii.1	. 2 ⊢ (𝜑 ↔ 𝜓)
2		notbi 670	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sylnbi  682  xchnxbi  684  xchbinx  686  nndc  856  xorcom  1430  xordidc  1441  dcfromnotnotr  1490  dcfromcon  1491  sbn  2003  neirr  2409  dfrex2dc  2521  ddifstab  3336  dfss4st  3437  ssddif  3438  difin  3441  difundi  3456  difindiss  3458  indifdir  3460  rabeq0  3521  abeq0  3522  snprc  3731  difprsnss  3806  uni0b  3913  disjnim  4073  brdif  4137  unidif0  4251  dtruex  4651  dcextest  4673  difopab  4855  cnvdif  5135  imadiflem  5400  imadif  5401  brprcneu  5622  poxp  6384  finexdc  7073  snexxph  7128  infmoti  7206  ismkvnex  7333  pw1nel3  7427  onntri35  7433  netap  7451  prltlu  7685  recexprlemdisj  7828  axpre-apti  8083  dfinfre  9114  fzdifsuc  10289  fzp1nel  10312  swrdccatin2  11276  ntreq0  14821  bj-nnor  16153  bj-nndcALT  16177  nnti  16415