Theorem esumeq2dv

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017)

	Ref	Expression
Hypothesis	esumeq2dv.1	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
Assertion	esumeq2dv	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$

Proof

Step	Hyp	Ref	Expression
1		esumeq2dv.1	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
2		nfv	$$⊢Ⅎk\phi$$
3	1	ralrimiva	$$⊢\phi \to \forall k\in AB=C$$
4	2 3	esumeq2d	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$