Theorem prodeq2dv

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

	Ref	Expression
Hypothesis	prodeq2dv.1	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
Assertion	prodeq2dv	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		prodeq2dv.1	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
2	1	ralrimiva	$$⊢\phi \to \forall k\in AB=C$$
3	2	prodeq2d	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$