Theorem nfdif

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003) (Revised by Mario Carneiro, 13-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 14-May-2025)

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

Proof

Step	Hyp	Ref	Expression
1		nfdif.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfdif.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		eldif	$$⊢y\in \left(A\setminus B\right)\leftrightarrow \left(y\in A\wedge \neg y\in B\right)$$
4	1	nfcri	$$⊢Ⅎxy\in A$$
5	2	nfcri	$$⊢Ⅎxy\in B$$
6	5	nfn	$$⊢Ⅎx\neg y\in B$$
7	4 6	nfan	$$⊢Ⅎx\left(y\in A\wedge \neg y\in B\right)$$
8	3 7	nfxfr	$$⊢Ⅎxy\in \left(A\setminus B\right)$$
9	8	nfci	$$⊢\underset{\_}{Ⅎ}x\left(A\setminus B\right)$$