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Metamath Proof Explorer
Description: Equality deduction for sum. Note that unlike sumeq2dv , k may
occur in ph . (Contributed by NM, 1-Nov-2005)
|
|
Ref |
Expression |
|
Hypothesis |
sumeq2d.1 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |
|
Assertion |
sumeq2d |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
sumeq2d.1 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |
| 2 |
|
sumeq2 |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 ) |