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	⊢ Ⅎ 𝑥 𝐴
	nfdif.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfdif	⊢ Ⅎ 𝑥 ( 𝐴 ∖ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfdif.1	⊢ Ⅎ 𝑥 𝐴
2		nfdif.2	⊢ Ⅎ 𝑥 𝐵
3		eldif	⊢ ( 𝑦 ∈ ( 𝐴 ∖ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵 ) )
4	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
5	2	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐵
6	5	nfn	⊢ Ⅎ 𝑥 ¬ 𝑦 ∈ 𝐵
7	4 6	nfan	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵 )
8	3 7	nfxfr	⊢ Ⅎ 𝑥 𝑦 ∈ ( 𝐴 ∖ 𝐵 )
9	8	nfci	⊢ Ⅎ 𝑥 ( 𝐴 ∖ 𝐵 )