Theorem cbviunvw2

Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbviunvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
	cbviunvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
Assertion	cbviunvw2	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷

Proof

Step	Hyp	Ref	Expression
1		cbviunvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
2		cbviunvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
3	1	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) )
4	2 3	cbvrexvw2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃ 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 )
5	4	abbii	⊢ { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 } = { 𝑡 ∣ ∃ 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 }
6		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 }
7		df-iun	⊢ ∪ 𝑦 ∈ 𝐵 𝐷 = { 𝑡 ∣ ∃ 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 }
8	5 6 7	3eqtr4i	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷