Theorem cbvprodi

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

	Ref	Expression
Hypotheses	cbvprodi.1	⊢ Ⅎ 𝑘 𝐵
	cbvprodi.2	⊢ Ⅎ 𝑗 𝐶
	cbvprodi.3	⊢ ( 𝑗 = 𝑘 → 𝐵 = 𝐶 )
Assertion	cbvprodi	⊢ ∏ 𝑗 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶

Proof

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