Theorem nesym 2381

Description: Characterization of inequality in terms of reversed equality (see bicom 139). (Contributed by BJ, 7-Jul-2018.)

Assertion

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

Proof of Theorem nesym

Step	Hyp	Ref	Expression
1		eqcom 2167	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2	1	necon3abii 2372	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 104   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2337

This theorem is referenced by:  nesymi  2382  nesymir  2383  0neqopab  5887  fzdifsuc  10016  isprm3  12050