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Theorem sbcieg 3023
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
Hypothesis
Ref Expression
sbcieg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbcieg  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem sbcieg
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 nfv 1605 . . 3  |-  F/ x ps
3 sbcieg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3sbciegf 3022 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  ps ) )
51, 4syl 15 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  sbcie  3025  ralsng  3672  rexsng  3673  ralrnmpt  5669  fpwwe2lem3  8255  nn1suc  9767  fgcl  17573  cfinfil  17588  csdfil  17589  supfil  17590  fin1aufil  17627  ifeqeqx  23034  2nn0ind  27030  zindbi  27031  trsbc  28304  onfrALTlem5  28307  trsbcVD  28653  onfrALTlem5VD  28661  bnj1452  29082  cdlemk35s  31126  cdlemk39s  31128  cdlemk42  31130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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