Theorem cbvsumvw2

Description: Change bound variable and the set of integers in a sum, using implicit substitution. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	cbvsumvw2.1	⊢ 𝐴 = 𝐵
	cbvsumvw2.2	⊢ ( 𝑗 = 𝑘 → 𝐶 = 𝐷 )
Assertion	cbvsumvw2	⊢ Σ 𝑗 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		cbvsumvw2.1	⊢ 𝐴 = 𝐵
2		cbvsumvw2.2	⊢ ( 𝑗 = 𝑘 → 𝐶 = 𝐷 )
3	2	cbvsumv	⊢ Σ 𝑗 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐴 𝐷
4	1	sumeq1i	⊢ Σ 𝑘 ∈ 𝐴 𝐷 = Σ 𝑘 ∈ 𝐵 𝐷
5	3 4	eqtri	⊢ Σ 𝑗 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷