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Theorem abbi1dv 2399
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 1617 . 2  |-  ( ph  ->  A. x ( ps  <->  x  e.  A ) )
3 abeq1 2389 . 2  |-  ( { x  |  ps }  =  A  <->  A. x ( ps  <->  x  e.  A ) )
42, 3sylibr 203 1  |-  ( ph  ->  { x  |  ps }  =  A )
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:  abidnf  2934  csbtt  3093  csbvarg  3108  csbie2g  3127  abvor0  3472  iinxsng  3978  enfin2i  7947  fin1a2lem11  8036  hashf1  11395  shftuz  11564  psrbaglefi  16118  vmappw  20354  predep  24192  dffprod  25319  grpodivfo  25374  trran2  25393  ltrran2  25403  rltrran  25414  trran  25614  hdmap1fval  31987  hdmapfval  32020  hgmapfval  32079
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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