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Theorem nfrabxy 2550
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 2369 . 2  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
2 nfrabxy.2 . . . . 5  |-  F/_ x A
32nfcri 2223 . . . 4  |-  F/ x  y  e.  A
4 nfrabxy.1 . . . 4  |-  F/ x ph
53, 4nfan 1503 . . 3  |-  F/ x
( y  e.  A  /\  ph )
65nfab 2234 . 2  |-  F/_ x { y  |  ( y  e.  A  /\  ph ) }
71, 6nfcxfr 2226 1  |-  F/_ x { y  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1395    e. wcel 1439   {cab 2075   F/_wnfc 2216   {crab 2364
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369
This theorem is referenced by:  nfdif  3124  nfin  3209  nfse  4179  mpt2xopoveq  6021  nfsup  6743  caucvgprprlemaddq  7330
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