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| Mirrors > Home > MPE Home > Th. List > neeq12d | Structured version Visualization version GIF version | ||
| Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
| Ref | Expression |
|---|---|
| neeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| neeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| neeq12d | ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | neeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 3 | 1, 2 | eqeq12d 2753 | . 2 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| 4 | 3 | necon3bid 2985 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) |
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