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Theorem abbi2dv 2276
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 1854 . 2  |-  ( ph  ->  A. x ( x  e.  A  <->  ps )
)
3 abeq2 2266 . 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 1333    = wceq 1335    e. wcel 2128   {cab 2143
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153
This theorem is referenced by:  sbab  2285  iftrue  3510  iffalse  3513  iniseg  4957  fncnvima2  5587  isoini  5765  dftpos3  6206  unfiexmid  6859  tgval3  12445  txrest  12663  cnblcld  12922
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