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Theorem nfabd 2247
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
nfabd.1  |-  F/ y
ph
nfabd.2  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfabd  |-  ( ph  -> 
F/_ x { y  |  ps } )

Proof of Theorem nfabd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1466 . 2  |-  F/ z
ph
2 df-clab 2075 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
3 nfabd.1 . . . 4  |-  F/ y
ph
4 nfabd.2 . . . 4  |-  ( ph  ->  F/ x ps )
53, 4nfsbd 1899 . . 3  |-  ( ph  ->  F/ x [ z  /  y ] ps )
62, 5nfxfrd 1409 . 2  |-  ( ph  ->  F/ x  z  e. 
{ y  |  ps } )
71, 6nfcd 2223 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1394    e. wcel 1438   [wsb 1692   {cab 2074   F/_wnfc 2215
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-nfc 2217
This theorem is referenced by:  nfsbcd  2859  nfcsb1d  2961  nfcsbd  2964  nfifd  3416  nfunid  3658  nfiotadxy  4978
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