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Theorem abbi2dv 2398
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 1617 . 2  |-  ( ph  ->  A. x ( x  e.  A  <->  ps )
)
3 abeq2 2388 . 2  |-  ( A  =  { x  |  ps }  <->  A. x
( x  e.  A  <->  ps ) )
42, 3sylibr 203 1  |-  ( ph  ->  A  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269
This theorem is referenced by:  sbab  2405  iftrue  3571  iffalse  3572  dfopif  3793  iniseg  5044  fncnvima2  5647  isoini  5835  dftpos3  6252  hartogslem1  7257  r1val2  7509  cardval2  7624  dfac3  7748  wrdval  11416  submacs  14442  dfrhm2  15498  lsppr  15846  rspsn  16006  znunithash  16518  tgval3  16701  txrest  17325  xkoptsub  17348  cnblcld  18284  shft2rab  18867  sca2rab  18871  grpoinvf  20907  elpjrn  22770  setlikespec  24187  prj1b  25079  prj3  25080  repcpwti  25161  nZdef  25180  neibastop3  26311  lkrval2  29280  lshpset2N  29309  hdmapoc  32124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279
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