Theorem sumeq2d

Description: Equality deduction for sum. Note that unlike sumeq2dv , k may occur in ph . (Contributed by NM, 1-Nov-2005)

	Ref	Expression
Hypothesis	sumeq2d.1	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
Assertion	sumeq2d	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sumeq2d.1	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
2		sumeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 𝐶 )