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Theorem nfin 3143
 Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfin.1 𝑥𝐴
nfin.2 𝑥𝐵
Assertion
Ref Expression
nfin 𝑥(𝐴𝐵)

Proof of Theorem nfin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfin5 2925 . 2 (𝐴𝐵) = {𝑦𝐴𝑦𝐵}
2 nfin.2 . . . 4 𝑥𝐵
32nfcri 2172 . . 3 𝑥 𝑦𝐵
4 nfin.1 . . 3 𝑥𝐴
53, 4nfrabxy 2490 . 2 𝑥{𝑦𝐴𝑦𝐵}
61, 5nfcxfr 2175 1 𝑥(𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Ⅎwnfc 2165  {crab 2310   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-in 2924 This theorem is referenced by:  csbing  3144  nfres  4614
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