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| Mirrors > Home > MPE Home > Th. List > 3eqtr4a | Structured version Visualization version GIF version | ||
| Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
| Ref | Expression |
|---|---|
| 3eqtr4a.1 | ⊢ 𝐴 = 𝐵 |
| 3eqtr4a.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
| 3eqtr4a.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
| Ref | Expression |
|---|---|
| 3eqtr4a | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3eqtr4a.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐴) | |
| 2 | 3eqtr4a.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 3 | 1, 2 | eqtrdi 2793 | . 2 ⊢ (𝜑 → 𝐶 = 𝐵) |
| 4 | 3eqtr4a.3 | . 2 ⊢ (𝜑 → 𝐷 = 𝐵) | |
| 5 | 3, 4 | eqtr4d 2780 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
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