Theorem esumeq1

Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017)

	Ref	Expression
Assertion	esumeq1	⊢ ( 𝐴 = 𝐵 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2		eqidd	⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐶 )
3	1 2	esumeq12d	⊢ ( 𝐴 = 𝐵 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐶 )