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Theorem abbi1dv 2592
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Hypothesis
Ref Expression
abbi1dv.1  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
Assertion
Ref Expression
abbi1dv  |-  ( ph  ->  { x  |  ps }  =  A )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem abbi1dv
StepHypRef Expression
1 abbi1dv.1 . . . 4  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
21bicomd 201 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
32abbi2dv 2591 . 2  |-  ( ph  ->  A  =  { x  |  ps } )
43eqcomd 2462 1  |-  ( ph  ->  { x  |  ps }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449
This theorem is referenced by:  abidnf  3235  csbtt  3409  csbie2g  3429  abvor0  3766  csbvarg  3811  iinxsng  4358  enfin2i  8605  fin1a2lem11  8694  hashf1  12332  shftuz  12680  psrbaglefi  17574  psrbaglefiOLD  17575  vmappw  22597  predep  27820  hdmap1fval  35805  hdmapfval  35838  hgmapfval  35897
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