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Mirrors > Home > MPE Home > Th. List > sumeq2sdv | Structured version Visualization version GIF version |
Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) |
Ref | Expression |
---|---|
sumeq2sdv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sumeq2sdv | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumeq2sdv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | sumeq2dv 14281 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶) |
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