Theorem esumeq2sdv

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016)

	Ref	Expression
Hypothesis	esumeq2sdv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	esumeq2sdv	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		esumeq2sdv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	esumeq2dv	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )