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| Description: Inference associated with df-ne 2941. (Contributed by BJ, 7-Jul-2018.) | 
| Ref | Expression | 
|---|---|
| neir.1 | ⊢ ¬ 𝐴 = 𝐵 | 
| Ref | Expression | 
|---|---|
| neir | ⊢ 𝐴 ≠ 𝐵 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | neir.1 | . 2 ⊢ ¬ 𝐴 = 𝐵 | |
| 2 | df-ne 2941 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ 𝐴 ≠ 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ≠ wne 2940 | 
| 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 207 df-ne 2941 | 
| This theorem is referenced by: nesymir 2999 vn0 4345 nsuceq0 6467 onnev 6511 nlim2 8528 ax1ne0 11200 ine0 11698 nosgnn0i 27704 1nei 32747 bj-pinftynminfty 37228 | 
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