Theorem sumeq2i

Description: Equality inference for sum. (Contributed by NM, 3-Dec-2005)

	Ref	Expression
Hypothesis	sumeq2i.1	⊢ ( 𝑘 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion	sumeq2i	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		sumeq2i.1	⊢ ( 𝑘 ∈ 𝐴 → 𝐵 = 𝐶 )
2		sumeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )
3	2 1	mprg	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶