Theorem cbviundavw2

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

	Ref	Expression
Hypotheses	cbviundavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
	cbviundavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbviundavw2	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		cbviundavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
2		cbviundavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3	1	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) )
4	3 2	cbvrexdva2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃ 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 ) )
5	4	abbidv	⊢ ( 𝜑 → { 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	3eqtr4g	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 )