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Theorem abbi2dv 2502
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbirdv.1  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
Assertion
Ref Expression
abbi2dv  |-  ( ph  ->  A  =  { x  |  ps } )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem abbi2dv
StepHypRef Expression
1 abbirdv.1 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
21alrimiv 1638 . 2  |-  ( ph  ->  A. x ( x  e.  A  <->  ps )
)
3 abeq2 2492 . 2  |-  ( A  =  { x  |  ps }  <->  A. x
( x  e.  A  <->  ps ) )
42, 3sylibr 204 1  |-  ( ph  ->  A  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2373
This theorem is referenced by:  sbab  2509  iftrue  3688  iffalse  3689  dfopif  3923  iniseg  5175  fncnvima2  5791  isoini  5997  dftpos3  6433  hartogslem1  7444  r1val2  7696  cardval2  7811  dfac3  7935  wrdval  11657  submacs  14692  dfrhm2  15748  lsppr  16092  rspsn  16252  znunithash  16768  tgval3  16951  txrest  17584  xkoptsub  17607  cnextf  18018  cnblcld  18680  shft2rab  19271  sca2rab  19275  grpoinvf  21676  elpjrn  23541  ofrn2  23896  setlikespec  25211  neibastop3  26082  lkrval2  29205  lshpset2N  29234  hdmapoc  32049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383
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