Theorem prodeq12dv

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

	Ref	Expression
Hypotheses	prodeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	prodeq12dv.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐷 )
Assertion	prodeq12dv	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		prodeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		prodeq12dv.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐷 )
3	2	prodeq2dv	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐴 𝐷 )
4	1	prodeq1d	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐷 = ∏ 𝑘 ∈ 𝐵 𝐷 )
5	3 4	eqtrd	⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷 )