Theorem prodeq12dv

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

	Ref	Expression
Hypotheses	prodeq12dv.1	$$⊢\phi \to A=B$$
	prodeq12dv.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
Assertion	prodeq12dv	$$⊢\phi \to \prod _{k\in A}C=\prod _{k\in B}D$$

Proof

Step	Hyp	Ref	Expression
1		prodeq12dv.1	$$⊢\phi \to A=B$$
2		prodeq12dv.2	$$⊢\left(\phi \wedge k\in A\right)\to C=D$$
3	2	prodeq2dv	$$⊢\phi \to \prod _{k\in A}C=\prod _{k\in A}D$$
4	1	prodeq1d	$$⊢\phi \to \prod _{k\in A}D=\prod _{k\in B}D$$
5	3 4	eqtrd	$$⊢\phi \to \prod _{k\in A}C=\prod _{k\in B}D$$