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Theorem nfrabxy 2561
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 2379 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nfrabxy.2 . . . . 5 𝑥𝐴
32nfcri 2229 . . . 4 𝑥 𝑦𝐴
4 nfrabxy.1 . . . 4 𝑥𝜑
53, 4nfan 1509 . . 3 𝑥(𝑦𝐴𝜑)
65nfab 2240 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
71, 6nfcxfr 2232 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 103  wnf 1401  wcel 1445  {cab 2081  wnfc 2222  {crab 2374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rab 2379
This theorem is referenced by:  nfdif  3136  nfin  3221  nfse  4192  elfvmptrab1  5433  mpt2xopoveq  6043  nfsup  6767  caucvgprprlemaddq  7364
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