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Theorem nfabd 2212
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 1437 . 2  |-  F/ z
ph
2 df-clab 2043 . . 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 1867 . . 3  |-  ( ph  ->  F/ x [ z  /  y ] ps )
62, 5nfxfrd 1380 . 2  |-  ( ph  ->  F/ x  z  e. 
{ y  |  ps } )
71, 6nfcd 2189 1  |-  ( ph  -> 
F/_ x { y  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1365    e. wcel 1409   [wsb 1661   {cab 2042   F/_wnfc 2181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-nfc 2183
This theorem is referenced by:  nfsbcd  2806  nfcsb1d  2908  nfcsbd  2911  nfifd  3383  nfunid  3615  nfiotadxy  4898
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