Theorem neir 2963

Description: Inference associated with df-ne 2961. (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 2961	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 234	1 ⊢ 𝐴 ≠ 𝐵

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ≠ wne 2960

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2961

This theorem is used by:  nesymir  3018  vn0  4298  vn0OLD  4299  nsuceq0  6450  onnev  6493  nlim2  8477  ax1ne0  11156  ine0  11660  nosgnn0i  27854  1nei  33128  bj-vn0ALT  37741  bj-pinftynminfty  37904