Metamath Proof Explorer


Theorem nfin

Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 14-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 14-May-2025)

Ref Expression
Hypotheses nfin.1
|- F/_ x A
nfin.2
|- F/_ x B
Assertion nfin
|- F/_ x ( A i^i B )

Proof

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