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Theorem abidnf 3272
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 1808 . . 3  |-  ( A. x  z  e.  A  ->  z  e.  A )
2 nfcr 2620 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
32nfrd 1823 . . 3  |-  ( F/_ x A  ->  ( z  e.  A  ->  A. x  z  e.  A )
)
41, 3impbid2 204 . 2  |-  ( F/_ x A  ->  ( A. x  z  e.  A  <->  z  e.  A ) )
54abbi1dv 2605 1  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   F/_wnfc 2615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617
This theorem is referenced by:  dedhb  3273  nfopd  4230  nfimad  5346  nffvd  5875
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