Theorem prodeq1d

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

	Ref	Expression
Hypothesis	prodeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	prodeq1d	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		prodeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		prodeq1	⊢ ( 𝐴 = 𝐵 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐶 )