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Theorem abbi1dv 2257
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 1846 . 2  |-  ( ph  ->  A. x ( ps  <->  x  e.  A ) )
3 abeq1 2247 . 2  |-  ( { x  |  ps }  =  A  <->  A. x ( ps  <->  x  e.  A ) )
42, 3sylibr 133 1  |-  ( ph  ->  { x  |  ps }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2123
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133
This theorem is referenced by:  abidnf  2847  csbtt  3009  csbvarg  3025  csbie2g  3045  abvor0dc  3381  iinxsng  3881  shftuz  10582
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