Theorem prodeq1d

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

	Ref	Expression
Hypothesis	prodeq1d.1	$$⊢\phi \to A=B$$
Assertion	prodeq1d	$$⊢\phi \to \prod _{k\in A}C=\prod _{k\in B}C$$

Proof

Step	Hyp	Ref	Expression
1		prodeq1d.1	$$⊢\phi \to A=B$$
2		prodeq1	$$⊢A=B\to \prod _{k\in A}C=\prod _{k\in B}C$$
3	1 2	syl	$$⊢\phi \to \prod _{k\in A}C=\prod _{k\in B}C$$