Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006) (Proof shortened by Glauco Siliprandi, 5-Apr-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | sumeq2sdv.1 | ⊢ ( 𝜑 → 𝐵 = 𝐶 ) | |
| Assertion | sumeq2sdv | ⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sumeq2sdv.1 | ⊢ ( 𝜑 → 𝐵 = 𝐶 ) | |
| 2 | 1 | ralrimivw | ⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |
| 3 | 2 | sumeq2d | ⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 ) |