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	$$⊢\phi \to \forall k\in AB=C$$
Assertion	sumeq2d	$$⊢\phi \to \sum _{k\in A}B=\sum _{k\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		sumeq2d.1	$$⊢\phi \to \forall k\in AB=C$$
2		sumeq2	$$⊢\forall k\in AB=C\to \sum _{k\in A}B=\sum _{k\in A}C$$
3	1 2	syl	$$⊢\phi \to \sum _{k\in A}B=\sum _{k\in A}C$$