Theorem sumeq12dv

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

	Ref	Expression
Hypotheses	sumeq12dv.1	$$⊢\phi \to A=B$$
	sumeq12dv.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
Assertion	sumeq12dv	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		sumeq12dv.1	$$⊢\phi \to A=B$$
2		sumeq12dv.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
3	2	sumeq2dv	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in A}D$$
4	1	sumeq1d	$$⊢\phi \to \sum _{k\in A}D=\sum _{k\in B}D$$
5	3 4	eqtrd	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in B}D$$