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	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
Assertion	sumeq2dv	$$⊢\phi \to \sum _{k\in A}B=\sum _{k\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		sumeq2dv.1	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
2	1	ralrimiva	$$⊢\phi \to \forall k\in AB=C$$
3	2	sumeq2d	$$⊢\phi \to \sum _{k\in A}B=\sum _{k\in A}C$$