Theorem prodeq12dv

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

	Ref	Expression
Hypotheses	prodeq12dv.1	|- ( ph -> A = B )
	prodeq12dv.2	|- ( ( ph /\ k e. A ) -> C = D )
Assertion	prodeq12dv	|- ( ph -> prod_ k e. A C = prod_ k e. B D )

Proof

Step	Hyp	Ref	Expression
1		prodeq12dv.1	|- ( ph -> A = B )
2		prodeq12dv.2	|- ( ( ph /\ k e. A ) -> C = D )
3	2	prodeq2dv	|- ( ph -> prod_ k e. A C = prod_ k e. A D )
4	1	prodeq1d	|- ( ph -> prod_ k e. A D = prod_ k e. B D )
5	3 4	eqtrd	|- ( ph -> prod_ k e. A C = prod_ k e. B D )