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Theorem abbi2dv 2289
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 1867 . 2  |-  ( ph  ->  A. x ( x  e.  A  <->  ps )
)
3 abeq2 2279 . 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 1346    = wceq 1348    e. wcel 2141   {cab 2156
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166
This theorem is referenced by:  sbab  2298  iftrue  3531  iffalse  3534  iniseg  4983  fncnvima2  5617  isoini  5797  dftpos3  6241  unfiexmid  6895  tgval3  12852  txrest  13070  cnblcld  13329
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