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	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
Assertion	prodeq2d	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		prodeq2d.1	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
2		prodeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )