Theorem esumeq12d

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017)

	Ref	Expression
Hypotheses	esumeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	esumeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	esumeq12d	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		esumeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		esumeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	2	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 = 𝐷 )
4	1 3	esumeq12dva	⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐶 = Σ* 𝑘 ∈ 𝐵 𝐷 )