Theorem nfdju

Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022)

	Ref	Expression
Hypotheses	nfdju.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfdju.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfdju	$$⊢\underset{\_}{Ⅎ}x\left(A\mathrm{\bigsqcup ︀}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfdju.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfdju.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		df-dju	$$⊢\left(A\mathrm{\bigsqcup ︀}B\right)=\left(\left\{\varnothing \right\}\times A\right)\cup \left(\left\{1_{𝑜}\right\}\times B\right)$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}x\left\{\varnothing \right\}$$
5	4 1	nfxp	$$⊢\underset{\_}{Ⅎ}x\left(\left\{\varnothing \right\}\times A\right)$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}x\left\{1_{𝑜}\right\}$$
7	6 2	nfxp	$$⊢\underset{\_}{Ⅎ}x\left(\left\{1_{𝑜}\right\}\times B\right)$$
8	5 7	nfun	$$⊢\underset{\_}{Ⅎ}x\left(\left(\left\{\varnothing \right\}\times A\right)\cup \left(\left\{1_{𝑜}\right\}\times B\right)\right)$$
9	3 8	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left(A\mathrm{\bigsqcup ︀}B\right)$$