Theorem nesym 2465

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 2240	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2	1	necon3abii 2456	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐵 = 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 105   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  nesymi  2466  nesymir  2467  0neqopab  6133  fzdifsuc  10490  isprm3  12898