Theorem cbvprodv

Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017)

	Ref	Expression
Hypothesis	cbvprod.1	⊢ ( 𝑗 = 𝑘 → 𝐵 = 𝐶 )
Assertion	cbvprodv	⊢ ∏ 𝑗 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		cbvprod.1	⊢ ( 𝑗 = 𝑘 → 𝐵 = 𝐶 )
2		nfcv	⊢ Ⅎ 𝑘 𝐴
3		nfcv	⊢ Ⅎ 𝑗 𝐴
4		nfcv	⊢ Ⅎ 𝑘 𝐵
5		nfcv	⊢ Ⅎ 𝑗 𝐶
6	1 2 3 4 5	cbvprod	⊢ ∏ 𝑗 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶