Theorem esumeq1d

Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017)

	Ref	Expression
Hypotheses	esumeq1d.0	⊢ Ⅎ 𝑘 𝜑
	esumeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	esumeq1d	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		esumeq1d.0	⊢ Ⅎ 𝑘 𝜑
2		esumeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		eqidd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐶 )
4	1 2 3	esumeq12dvaf	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐶 )