Theorem prodeq2d

Description: Equality deduction for product. Note that unlike prodeq2dv , k may occur in ph . (Contributed by Scott Fenton, 4-Dec-2017)

	Ref	Expression
Hypothesis	prodeq2d.1	$$⊢\phi \to \forall k\in AB=C$$
Assertion	prodeq2d	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$

Proof

Step	Hyp	Ref	Expression
1		prodeq2d.1	$$⊢\phi \to \forall k\in AB=C$$
2		prodeq2	$$⊢\forall k\in AB=C\to \prod _{k\in A}B=\prod _{k\in A}C$$
3	1 2	syl	$$⊢\phi \to \prod _{k\in A}B=\prod _{k\in A}C$$