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Metamath Proof Explorer
Description: Equality deduction for product. Note that unlike prodeq2dv , k
may occur in ph . (Contributed by Scott Fenton, 4-Dec-2017)
|
|
Ref |
Expression |
|
Hypothesis |
prodeq2d.1 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |
|
Assertion |
prodeq2d |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
prodeq2d.1 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |
| 2 |
|
prodeq2 |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 ) |