Theorem esumeq2d

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016)

	Ref	Expression
Hypotheses	esumeq2d.0	$$⊢Ⅎk\phi$$
	esumeq2d.1	$$⊢\phi \to \forall k\in AB=C$$
Assertion	esumeq2d	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$

Proof

Step	Hyp	Ref	Expression
1		esumeq2d.0	$$⊢Ⅎk\phi$$
2		esumeq2d.1	$$⊢\phi \to \forall k\in AB=C$$
3		eqidd	$$⊢\phi \to A=A$$
4	2	r19.21bi	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
5	1 3 4	esumeq12dvaf	$$⊢\phi \to \underset{k\in A}{{\sum }^{*}}B=\underset{k\in A}{{\sum }^{*}}C$$