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Theorem abbi1dv 2199
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbildv.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 abbildv.1 . . 3  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
21alrimiv 1796 . 2  |-  ( ph  ->  A. x ( ps  <->  x  e.  A ) )
3 abeq1 2189 . 2  |-  ( { x  |  ps }  =  A  <->  A. x ( ps  <->  x  e.  A ) )
42, 3sylibr 132 1  |-  ( ph  ->  { x  |  ps }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434   {cab 2068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078
This theorem is referenced by:  abidnf  2761  csbtt  2919  csbvarg  2934  csbie2g  2953  abvor0dc  3276  iinxsng  3759  shftuz  9843
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