Theorem sumeq12dv

Description: Equality deduction for sum. (Contributed by NM, 1-Dec-2005)

	Ref	Expression
Hypotheses	sumeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sumeq12dv.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐷 )
Assertion	sumeq12dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		sumeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sumeq12dv.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐷 )
3	2	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐴 𝐷 )
4	1	sumeq1d	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐷 = Σ 𝑘 ∈ 𝐵 𝐷 )
5	3 4	eqtrd	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝐵 𝐷 )