Theorem esumeq12d

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

	Ref	Expression
Hypotheses	esumeq12d.1	$$⊢\phi \to A=B$$
	esumeq12d.2	$$⊢\phi \to C=D$$
Assertion	esumeq12d	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}C=\underset{k\in B}{{\sum }^{*}}D$$

Proof

Step	Hyp	Ref	Expression
1		esumeq12d.1	$$⊢\phi \to A=B$$
2		esumeq12d.2	$$⊢\phi \to C=D$$
3	2	adantr	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
4	1 3	esumeq12dva	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}C=\underset{k\in B}{{\sum }^{*}}D$$