Theorem prodeq2

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

	Ref	Expression
Assertion	prodeq2	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐵 = 𝐶 → ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) )
2	1	ralimi	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) )
3		prodeq2ii	⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )
4	2 3	syl	⊢ ( ∀ 𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )