Theorem esumeq2dv

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017)

	Ref	Expression
Hypothesis	esumeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	esumeq2dv	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		esumeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
2		nfv	⊢ Ⅎ 𝑘 𝜑
3	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
4	2 3	esumeq2d	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )