Theorem esumeq12dva

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017) (Revised by Thierry Arnoux, 29-Jun-2017)

	Ref	Expression
Hypotheses	esumeq12dva.1	$$⊢\phi \to A=B$$
	esumeq12dva.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
Assertion	esumeq12dva	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}C=\underset{k\in B}{{\sum }^{*}}D$$

Proof

Step	Hyp	Ref	Expression
1		esumeq12dva.1	$$⊢\phi \to A=B$$
2		esumeq12dva.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
3		nfv	$$⊢Ⅎk\phi$$
4	3 1 2	esumeq12dvaf	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}C=\underset{k\in B}{{\sum }^{*}}D$$