Theorem nfdif

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003) (Revised by Mario Carneiro, 13-Oct-2016)

	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		dfdif2	$$⊢A\setminus B=\left\{y\in A|\neg y\in B\right\}$$
4	2	nfcri	$$⊢Ⅎxy\in B$$
5	4	nfn	$$⊢Ⅎx\neg y\in B$$
6	5 1	nfrabw	$$⊢\underset{\_}{Ⅎ}x\left\{y\in A|\neg y\in B\right\}$$
7	3 6	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left(A\setminus B\right)$$