Theorem prodeq2dv

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

	Ref	Expression
Hypothesis	prodeq2dv.1	|- ( ( ph /\ k e. A ) -> B = C )
Assertion	prodeq2dv	|- ( ph -> prod_ k e. A B = prod_ k e. A C )

Proof

Step	Hyp	Ref	Expression
1		prodeq2dv.1	|- ( ( ph /\ k e. A ) -> B = C )
2	1	ralrimiva	|- ( ph -> A. k e. A B = C )
3	2	prodeq2d	|- ( ph -> prod_ k e. A B = prod_ k e. A C )