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| Mirrors > Home > MPE Home > Th. List > sumeq2dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| Ref | Expression |
|---|---|
| sumeq2dv.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| sumeq2dv | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sumeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶) |
| 3 | 2 | sumeq2d 15737 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
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