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Theorem abidnf 2769
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 1442 . . 3  |-  ( A. x  z  e.  A  ->  z  e.  A )
2 nfcr 2215 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
32nfrd 1454 . . 3  |-  ( F/_ x A  ->  ( z  e.  A  ->  A. x  z  e.  A )
)
41, 3impbid2 141 . 2  |-  ( F/_ x A  ->  ( A. x  z  e.  A  <->  z  e.  A ) )
54abbi1dv 2202 1  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2069   F/_wnfc 2210
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212
This theorem is referenced by:  dedhb  2770  nfopd  3607  nfimad  4727  nffvd  5239
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