Theorem prodeq2ad

Description: Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypothesis	prodeq2ad.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	prodeq2ad	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		prodeq2ad.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
2	1	ralrimivw	⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 )
3	2	prodeq2d	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )