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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 
|- F/_ x A
nfdif.2 
|- F/_ x B
Assertion nfdif 
|- F/_ x ( A \ B )

Proof

Step Hyp Ref Expression
1 nfdif.1 
 |-  F/_ x A
2 nfdif.2 
 |-  F/_ x B
3 eldif 
 |-  ( y e. ( A \ B ) <-> ( y e. A /\ -. y e. B ) )
4 1 nfcri 
 |-  F/ x y e. A
5 2 nfcri 
 |-  F/ x y e. B
6 5 nfn 
 |-  F/ x -. y e. B
7 4 6 nfan 
 |-  F/ x ( y e. A /\ -. y e. B )
8 3 7 nfxfr 
 |-  F/ x y e. ( A \ B )
9 8 nfci 
 |-  F/_ x ( A \ B )