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Theorem nfrabxy 2614
 Description: A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
Hypotheses
Ref Expression
nfrabxy.1 𝑥𝜑
nfrabxy.2 𝑥𝐴
Assertion
Ref Expression
nfrabxy 𝑥{𝑦𝐴𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrabxy
StepHypRef Expression
1 df-rab 2426 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nfrabxy.2 . . . . 5 𝑥𝐴
32nfcri 2276 . . . 4 𝑥 𝑦𝐴
4 nfrabxy.1 . . . 4 𝑥𝜑
53, 4nfan 1545 . . 3 𝑥(𝑦𝐴𝜑)
65nfab 2287 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
71, 6nfcxfr 2279 1 𝑥{𝑦𝐴𝜑}
 Colors of variables: wff set class Syntax hints:   ∧ wa 103  Ⅎwnf 1437   ∈ wcel 1481  {cab 2126  Ⅎwnfc 2269  {crab 2421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426 This theorem is referenced by:  nfdif  3201  nfin  3286  nfse  4270  elfvmptrab1  5522  mpoxopoveq  6144  nfsup  6886  caucvgprprlemaddq  7539  ctiunct  11987
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