Theorem neir 2961

Description: Inference associated with df-ne 2959. (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 2959	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 234	1 ⊢ 𝐴 ≠ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1570   ≠ wne 2958

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 210  df-ne 2959

This theorem is referenced by:  nesymir  3016  vn0  4298  vn0OLD  4299  nsuceq0  6446  onnev  6489  nlim2  8471  ax1ne0  11140  ine0  11644  nosgnn0i  27823  1nei  33082  bj-vn0ALT  37708  bj-pinftynminfty  37871