Theorem nfiung

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 . See nfiun for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfiung.1	$$⊢\underset{\_}{Ⅎ}yA$$
2		nfiung.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	nfrexg	$$⊢Ⅎy\exists x\in Az\in B$$
6	5	nfabg	$$⊢\underset{\_}{Ⅎ}y\left\{z|\exists x\in Az\in B\right\}$$
7	3 6	nfcxfr	$$⊢\underset{\_}{Ⅎ}y\bigcup _{x\in A}B$$