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

Proof

Step Hyp Ref Expression
1 nfin.1 𝑥 𝐴
2 nfin.2 𝑥 𝐵
3 elin ( 𝑦 ∈ ( 𝐴𝐵 ) ↔ ( 𝑦𝐴𝑦𝐵 ) )
4 1 nfcri 𝑥 𝑦𝐴
5 2 nfcri 𝑥 𝑦𝐵
6 4 5 nfan 𝑥 ( 𝑦𝐴𝑦𝐵 )
7 3 6 nfxfr 𝑥 𝑦 ∈ ( 𝐴𝐵 )
8 7 nfci 𝑥 ( 𝐴𝐵 )