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Theorem nfrabxy 2535
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  |-  F/ x ph
nfrabxy.2  |-  F/_ x A
Assertion
Ref Expression
nfrabxy  |-  F/_ x { y  e.  A  |  ph }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem nfrabxy
StepHypRef Expression
1 df-rab 2358 . 2  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
2 nfrabxy.2 . . . . 5  |-  F/_ x A
32nfcri 2214 . . . 4  |-  F/ x  y  e.  A
4 nfrabxy.1 . . . 4  |-  F/ x ph
53, 4nfan 1498 . . 3  |-  F/ x
( y  e.  A  /\  ph )
65nfab 2224 . 2  |-  F/_ x { y  |  ( y  e.  A  /\  ph ) }
71, 6nfcxfr 2217 1  |-  F/_ x { y  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102   F/wnf 1390    e. wcel 1434   {cab 2068   F/_wnfc 2207   {crab 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358
This theorem is referenced by:  nfdif  3094  nfin  3179  nfse  4104  mpt2xopoveq  5889  nfsup  6464  caucvgprprlemaddq  6960
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