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| Mirrors > Home > MPE Home > Th. List > 3eqtrrd | Structured version Visualization version GIF version | ||
| Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
| Ref | Expression |
|---|---|
| 3eqtrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| 3eqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
| 3eqtrd.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| 3eqtrrd | ⊢ (𝜑 → 𝐷 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3eqtrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 3eqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
| 3 | 1, 2 | eqtrd 2777 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
| 4 | 3eqtrd.3 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 5 | 3, 4 | eqtr2d 2778 | 1 ⊢ (𝜑 → 𝐷 = 𝐴) |
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