Metamath Proof Explorer


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 𝑥 𝐴
nfdif.2 𝑥 𝐵
Assertion nfdif 𝑥 ( 𝐴𝐵 )

Proof

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