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| Mirrors > Home > MPE Home > Th. List > pm2.61ine | Structured version Visualization version GIF version | ||
| Description: Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
| Ref | Expression |
|---|---|
| pm2.61ine.1 | ⊢ (𝐴 = 𝐵 → 𝜑) |
| pm2.61ine.2 | ⊢ (𝐴 ≠ 𝐵 → 𝜑) |
| Ref | Expression |
|---|---|
| pm2.61ine | ⊢ 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.61ine.2 | . 2 ⊢ (𝐴 ≠ 𝐵 → 𝜑) | |
| 2 | nne 2944 | . . 3 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
| 3 | pm2.61ine.1 | . . 3 ⊢ (𝐴 = 𝐵 → 𝜑) | |
| 4 | 2, 3 | sylbi 217 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 → 𝜑) |
| 5 | 1, 4 | pm2.61i 182 | 1 ⊢ 𝜑 |
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