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Theorem abidnf 2821
Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
abidnf  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Distinct variable groups:    x, z    z, A
Allowed substitution hint:    A( x)

Proof of Theorem abidnf
StepHypRef Expression
1 sp 1471 . . 3  |-  ( A. x  z  e.  A  ->  z  e.  A )
2 nfcr 2247 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
32nfrd 1483 . . 3  |-  ( F/_ x A  ->  ( z  e.  A  ->  A. x  z  e.  A )
)
41, 3impbid2 142 . 2  |-  ( F/_ x A  ->  ( A. x  z  e.  A  <->  z  e.  A ) )
54abbi1dv 2234 1  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    = wceq 1314    e. wcel 1463   {cab 2101   F/_wnfc 2242
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-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244
This theorem is referenced by:  dedhb  2822  nfopd  3688  nfimad  4848  nffvd  5387
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