Theorem prodeq2sdv

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

	Ref	Expression
Hypothesis	prodeq2sdv.1	$$⊢\phi \to B=C$$
Assertion	prodeq2sdv	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		prodeq2sdv.1	$$⊢\phi \to B=C$$
2	1	adantr	$$⊢\left(\phi \wedge k\in A\right)\to B=C$$
3	2	prodeq2dv	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$