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Theorem abbi1dv 2412
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 1621 . 2  |-  ( ph  ->  A. x ( ps  <->  x  e.  A ) )
3 abeq1 2402 . 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 1530    = wceq 1632    e. wcel 1696   {cab 2282
This theorem is referenced by:  abidnf  2947  csbtt  3106  csbvarg  3121  csbie2g  3140  abvor0  3485  iinxsng  3994  enfin2i  7963  fin1a2lem11  8052  hashf1  11411  shftuz  11580  psrbaglefi  16134  vmappw  20370  predep  24263  dffprod  25422  grpodivfo  25477  trran2  25496  ltrran2  25506  rltrran  25517  trran  25717  hdmap1fval  32609  hdmapfval  32642  hgmapfval  32701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292
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