Theorem sumeq2dv

Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006) (Revised by Mario Carneiro, 31-Jan-2014)

	Ref	Expression
Hypothesis	sumeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sumeq2dv.1	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 = 𝐶 )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
3	2	sumeq2d	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )