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Theorem nfrabxy 2507
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 2332 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nfrabxy.2 . . . . 5 𝑥𝐴
32nfcri 2188 . . . 4 𝑥 𝑦𝐴
4 nfrabxy.1 . . . 4 𝑥𝜑
53, 4nfan 1473 . . 3 𝑥(𝑦𝐴𝜑)
65nfab 2198 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
71, 6nfcxfr 2191 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 101  wnf 1365  wcel 1409  {cab 2042  wnfc 2181  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332
This theorem is referenced by:  nfdif  3093  nfin  3171  nfse  4106  mpt2xopoveq  5886  nfsup  6398  caucvgprprlemaddq  6864
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