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| Mirrors > Home > MPE Home > Th. List > eqnetrd | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
| Ref | Expression |
|---|---|
| eqnetrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqnetrd.2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
| Ref | Expression |
|---|---|
| eqnetrd | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqnetrd.2 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
| 2 | eqnetrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | 2 | neeq1d 3000 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
| 4 | 1, 3 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
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