Theorem cbviun

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006) (Revised by Andrew Salmon, 25-Jul-2011) Add disjoint variable condition to avoid ax-13 . See cbviung for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

	Ref	Expression
Hypotheses	cbviun.1	⊢ Ⅎ 𝑦 𝐵
	cbviun.2	⊢ Ⅎ 𝑥 𝐶
	cbviun.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
Assertion	cbviun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		cbviun.1	⊢ Ⅎ 𝑦 𝐵
2		cbviun.2	⊢ Ⅎ 𝑥 𝐶
3		cbviun.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
4	1	nfcri	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵
5	2	nfcri	⊢ Ⅎ 𝑥 𝑧 ∈ 𝐶
6	3	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) )
7	4 5 6	cbvrexw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 )
8	7	abbii	⊢ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 }
9		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
10		df-iun	⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 }
11	8 9 10	3eqtr4i	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶