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Theorem abbi1dv 2605
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Hypothesis
Ref Expression
abbi1dv.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 abbi1dv.1 . . . 4  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
21bicomd 201 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ps ) )
32abbi2dv 2604 . 2  |-  ( ph  ->  A  =  { x  |  ps } )
43eqcomd 2475 1  |-  ( ph  ->  { x  |  ps }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   {cab 2452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462
This theorem is referenced by:  abidnf  3277  csbtt  3451  csbie2g  3471  abvor0  3808  csbvarg  3853  iinxsng  4408  enfin2i  8713  fin1a2lem11  8802  hashf1  12487  shftuz  12882  psrbaglefi  17893  psrbaglefiOLD  17894  vmappw  23256  predep  29199  hdmap1fval  36995  hdmapfval  37028  hgmapfval  37087
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