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Theorem abbi2dv 2236
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 1830 . 2  |-  ( ph  ->  A. x ( x  e.  A  <->  ps )
)
3 abeq2 2226 . 2  |-  ( A  =  { x  |  ps }  <->  A. x
( x  e.  A  <->  ps ) )
42, 3sylibr 133 1  |-  ( ph  ->  A  =  { x  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113
This theorem is referenced by:  sbab  2244  iftrue  3449  iffalse  3452  iniseg  4881  fncnvima2  5509  isoini  5687  dftpos3  6127  unfiexmid  6774  tgval3  12154  txrest  12372  cnblcld  12631
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