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	|- ( ph -> A. k e. A B = C )
Assertion	prodeq2d	|- ( ph -> prod_ k e. A B = prod_ k e. A C )

Proof

Step	Hyp	Ref	Expression
1		prodeq2d.1	|- ( ph -> A. k e. A B = C )
2		prodeq2	|- ( A. k e. A B = C -> prod_ k e. A B = prod_ k e. A C )
3	1 2	syl	|- ( ph -> prod_ k e. A B = prod_ k e. A C )