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Theorem abidnf 3103
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 1763 . . 3  |-  ( A. x  z  e.  A  ->  z  e.  A )
2 nfcr 2564 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
32nfrd 1779 . . 3  |-  ( F/_ x A  ->  ( z  e.  A  ->  A. x  z  e.  A )
)
41, 3impbid2 196 . 2  |-  ( F/_ x A  ->  ( A. x  z  e.  A  <->  z  e.  A ) )
54abbi1dv 2552 1  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2422   F/_wnfc 2559
This theorem is referenced by:  dedhb  3104  nfopd  4001  nfimad  5212  nffvd  5737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561
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