Theorem neir 2948

Description: Inference associated with df-ne 2946. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
neir.1	⊢ ¬ 𝐴 = 𝐵

Assertion

Ref	Expression
neir	⊢ 𝐴 ≠ 𝐵

Proof of Theorem neir

Step	Hyp	Ref	Expression
1		neir.1	. 2 ⊢ ¬ 𝐴 = 𝐵
2		df-ne 2946	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 230	1 ⊢ 𝐴 ≠ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1542   ≠ wne 2945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-ne 2946

This theorem is referenced by:  nesymir  3004  vn0  4278  nsuceq0  6344  onnev  6385  ax1ne0  10915  ine0  11408  1nei  31065  nosgnn0i  33856  bj-pinftynminfty  35392