Theorem nesym 2995

Description: Characterization of inequality in terms of reversed equality (see bicom 222). (Contributed by BJ, 7-Jul-2018.)

Assertion

Ref	Expression
nesym	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)

Proof of Theorem nesym

Step	Hyp	Ref	Expression
1		eqcom 2742	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2	1	necon3abii 2985	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ≠ wne 2938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ne 2939

This theorem is referenced by:  ord1eln01  8533  ord2eln012  8534  fiming  9536  wemapsolem  9588  nn01to3  12981  xrltlen  13185  sgnn  15130  isprm3  16717  lspsncv0  21166  uvcvv0  21828  fvmptnn04if  22871  chfacfisf  22876  chfacfisfcpmat  22877  trfbas  23868  fbunfip  23893  trfil2  23911  iundisj2  25598  nosupbnd2lem1  27775  noinfbnd2lem1  27790  elnns2  28359  pthdlem2lem  29800  fusgr2wsp2nb  30363  iundisj2f  32610  iundisj2fi  32805  cvmscld  35258  poimirlem25  37632  hlrelat5N  39384  redvmptabs  42369  cmpfiiin  42685  gneispace  44124  iblcncfioo  45934  fourierdlem82  46144  elprneb  46979  fzopredsuc  47273  iccpartiltu  47347