Theorem nesym 2446

Description: Characterization of inequality in terms of reversed equality (see bicom 140). (Contributed by BJ, 7-Jul-2018.)

Assertion

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

Proof of Theorem nesym

Step	Hyp	Ref	Expression
1		eqcom 2232	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2	1	necon3abii 2437	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 105   = wceq 1397   ≠ wne 2401

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-cleq 2223  df-ne 2402

This theorem is referenced by:  nesymi  2447  nesymir  2448  0neqopab  6071  fzdifsuc  10321  isprm3  12713