Theorem esumeq2d

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016)

	Ref	Expression
Hypotheses	esumeq2d.0	⊢ Ⅎ 𝑘 𝜑
	esumeq2d.1	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
Assertion	esumeq2d	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		esumeq2d.0	⊢ Ⅎ 𝑘 𝜑
2		esumeq2d.1	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
3		eqidd	⊢ ( 𝜑 → 𝐴 = 𝐴 )
4	2	r19.21bi	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
5	1 3 4	esumeq12dvaf	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑘 ∈ 𝐴 𝐶 )