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Theorem nfabd 2319
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 1508 . 2  |-  F/ z
ph
2 df-clab 2144 . . 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 1957 . . 3  |-  ( ph  ->  F/ x [ z  /  y ] ps )
62, 5nfxfrd 1455 . 2  |-  ( ph  ->  F/ x  z  e. 
{ y  |  ps } )
71, 6nfcd 2294 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1440   [wsb 1742    e. wcel 2128   {cab 2143   F/_wnfc 2286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-nfc 2288
This theorem is referenced by:  nfsbcd  2956  nfcsb1d  3062  nfcsbd  3066  nfifd  3532  nfunid  3779  nfiotadw  5138  nfixpxy  6662
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