Theorem cbviunv

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003) Add disjoint variable condition to avoid ax-13 . See cbviunvg for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025)

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

Proof

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