Theorem nfiun

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014) Add disjoint variable condition to avoid ax-13 . See nfiung for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfiun.1	$$⊢\underset{\_}{Ⅎ}yA$$
	nfiun.2	$$⊢\underset{\_}{Ⅎ}yB$$
Assertion	nfiun	$$⊢\underset{\_}{Ⅎ}y\bigcup _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		nfiun.1	$$⊢\underset{\_}{Ⅎ}yA$$
2		nfiun.2	$$⊢\underset{\_}{Ⅎ}yB$$
3		df-iun	$$⊢\bigcup _{x\in A}B=\left\{z|\exists x\in Az\in B\right\}$$
4	2	nfcri	$$⊢Ⅎyz\in B$$
5	1 4	nfrexw	$$⊢Ⅎy\exists x\in Az\in B$$
6	5	nfab	$$⊢\underset{\_}{Ⅎ}y\left\{z|\exists x\in Az\in B\right\}$$
7	3 6	nfcxfr	$$⊢\underset{\_}{Ⅎ}y\bigcup _{x\in A}B$$