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Theorem nfrabw 2675
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
nfrabw.1 𝑥𝜑
nfrabw.2 𝑥𝐴
Assertion
Ref Expression
nfrabw 𝑥{𝑦𝐴𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrabw
StepHypRef Expression
1 df-rab 2481 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nfrabw.2 . . . . 5 𝑥𝐴
32nfcri 2330 . . . 4 𝑥 𝑦𝐴
4 nfrabw.1 . . . 4 𝑥𝜑
53, 4nfan 1576 . . 3 𝑥(𝑦𝐴𝜑)
65nfab 2341 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
71, 6nfcxfr 2333 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 104  wnf 1471  wcel 2164  {cab 2179  wnfc 2323  {crab 2476
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 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481
This theorem is referenced by:  nfdif  3280  nfin  3365  nfse  4370  elfvmptrab1  5644  elovmporab  6110  elovmporab1w  6111  mpoxopoveq  6284  nfsup  7041  caucvgprprlemaddq  7758  ctiunct  12587
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