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Theorem nfrabxy 2484
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{y Aφ}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)

Proof of Theorem nfrabxy
StepHypRef Expression
1 df-rab 2309 . 2 {y Aφ} = {y ∣ (y A φ)}
2 nfrabxy.2 . . . . 5 xA
32nfcri 2169 . . . 4 x y A
4 nfrabxy.1 . . . 4 xφ
53, 4nfan 1454 . . 3 x(y A φ)
65nfab 2179 . 2 x{y ∣ (y A φ)}
71, 6nfcxfr 2172 1 x{y Aφ}
Colors of variables: wff set class
Syntax hints:   wa 97  wnf 1346   wcel 1390  {cab 2023  wnfc 2162  {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309
This theorem is referenced by:  nfdif  3059  nfin  3137  nfse  4063  mpt2xopoveq  5796
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