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Theorem nfrabxy 2671
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 2477 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nfrabxy.2 . . . . 5 𝑥𝐴
32nfcri 2326 . . . 4 𝑥 𝑦𝐴
4 nfrabxy.1 . . . 4 𝑥𝜑
53, 4nfan 1576 . . 3 𝑥(𝑦𝐴𝜑)
65nfab 2337 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
71, 6nfcxfr 2329 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 104  wnf 1471  wcel 2160  {cab 2175  wnfc 2319  {crab 2472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477
This theorem is referenced by:  nfdif  3271  nfin  3356  nfse  4359  elfvmptrab1  5631  mpoxopoveq  6265  nfsup  7021  caucvgprprlemaddq  7737  ctiunct  12491
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