Theorem prodeq12i

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

	Ref	Expression
Hypotheses	prodeq12i.1	⊢ 𝐴 = 𝐵
	prodeq12i.2	⊢ ( 𝑘 ∈ 𝐴 → 𝐶 = 𝐷 )
Assertion	prodeq12i	⊢ ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		prodeq12i.1	⊢ 𝐴 = 𝐵
2		prodeq12i.2	⊢ ( 𝑘 ∈ 𝐴 → 𝐶 = 𝐷 )
3	2	prodeq2i	⊢ ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐴 𝐷
4	1	prodeq1i	⊢ ∏ 𝑘 ∈ 𝐴 𝐷 = ∏ 𝑘 ∈ 𝐵 𝐷
5	3 4	eqtri	⊢ ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷