Theorem neir 2958

Description: Inference associated with df-ne 2956. (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 2956	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 234	1 ⊢ 𝐴 ≠ 𝐵

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ≠ wne 2955

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 2956

This theorem is used by:  nesymir  3013  vn0  4291  vn0OLD  4292  nsuceq0  6443  onnev  6486  nlim2  8477  ax1ne0  11169  ine0  11673  nosgnn0i  27895  1nei  33208  bj-vn0ALT  37816  bj-pinftynminfty  37979