Theorem sumeq12sdv

Description: Equality deduction for sum. General version of sumeq2sdv . (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	sumeq12sdv.1	$$⊢\phi \to A=B$$
	sumeq12sdv.2	$$⊢\phi \to C=D$$
Assertion	sumeq12sdv	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		sumeq12sdv.1	$$⊢\phi \to A=B$$
2		sumeq12sdv.2	$$⊢\phi \to C=D$$
3	1	sumeq1d	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in B}C$$
4	2	sumeq2sdv	$$⊢\phi \to \sum _{k\in B}C=\sum _{k\in B}D$$
5	3 4	eqtrd	$$⊢\phi \to \sum _{k\in A}C=\sum _{k\in B}D$$