Theorem cbviunf

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)

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

Proof

Step	Hyp	Ref	Expression
1		cbviunf.x	⊢ Ⅎ 𝑥 𝐴
2		cbviunf.y	⊢ Ⅎ 𝑦 𝐴
3		cbviunf.1	⊢ Ⅎ 𝑦 𝐵
4		cbviunf.2	⊢ Ⅎ 𝑥 𝐶
5		cbviunf.3	⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 )
6	3	nfcri	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵
7	4	nfcri	⊢ Ⅎ 𝑥 𝑧 ∈ 𝐶
8	5	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) )
9	1 2 6 7 8	cbvrexfw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 )
10	9	abbii	⊢ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 }
11		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
12		df-iun	⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = { 𝑧 ∣ ∃ 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 }
13	10 11 12	3eqtr4i	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶