Theorem prodeq12i

Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017)

	Ref	Expression
Hypotheses	prodeq12i.1	$$⊢A=B$$
	prodeq12i.2	$$⊢k\in A\to C=D$$
Assertion	prodeq12i	$$⊢\prod _{k\in A}C=\prod _{k\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		prodeq12i.1	$$⊢A=B$$
2		prodeq12i.2	$$⊢k\in A\to C=D$$
3	2	prodeq2i	$$⊢\prod _{k\in A}C=\prod _{k\in A}D$$
4	1	prodeq1i	$$⊢\prod _{k\in A}D=\prod _{k\in B}D$$
5	3 4	eqtri	$$⊢\prod _{k\in A}C=\prod _{k\in B}D$$