Theorem cbviundavw

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

	Ref	Expression
Hypothesis	cbviundavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion	cbviundavw	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		cbviundavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
2	1	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶 ) )
3	2	cbvrexdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∃ 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 ) )
4	3	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 } = { 𝑡 ∣ ∃ 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 } )
5		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 }
6		df-iun	⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∃ 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 }
7	4 5 6	3eqtr4g	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 )