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| Mirrors > Home > MPE Home > Th. List > eqneqall | Structured version Visualization version GIF version | ||
| Description: A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
| Ref | Expression |
|---|---|
| eqneqall | ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ne 2941 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
| 2 | pm2.24 124 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝜑)) | |
| 3 | 1, 2 | biimtrid 242 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) |
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