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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 )