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Theorem abidnf 2947
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 1728 . . 3  |-  ( A. x  z  e.  A  ->  z  e.  A )
2 nfcr 2424 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
32nfrd 1755 . . 3  |-  ( F/_ x A  ->  ( z  e.  A  ->  A. x  z  e.  A )
)
41, 3impbid2 195 . 2  |-  ( F/_ x A  ->  ( A. x  z  e.  A  <->  z  e.  A ) )
54abbi1dv 2412 1  |-  ( F/_ x A  ->  { z  |  A. x  z  e.  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419
This theorem is referenced by:  dedhb  2948  nfopd  3829  nfimad  5037  nffvd  5550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421
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