Theorem neir 2343

Description: Inference associated with df-ne 2341. (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 2341	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	mpbir 145	1 ⊢ 𝐴 ≠ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  exmidonfinlem  7170  pw1ne1  7206  pw1ne3  7207  sucpw1nel3  7210  ine0  8313  pwle2  14031