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Theorem sbcie 3159
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
Hypotheses
Ref Expression
sbcie.1  |-  A  e. 
_V
sbcie.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbcie  |-  ( [. A  /  x ]. ph  <->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem sbcie
StepHypRef Expression
1 sbcie.1 . 2  |-  A  e. 
_V
2 sbcie.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 3157 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  ps ) )
41, 3ax-mp 8 1  |-  ( [. A  /  x ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   _Vcvv 2920   [.wsbc 3125
This theorem is referenced by:  tfinds2  4806  findcard2  7311  ac6sfi  7314  ac6num  8319  fpwwe  8481  nn1suc  9981  wrdind  11750  cjth  11867  prmind2  13049  joinlem  14406  meetlem  14413  isghm  14965  islmod  15913  fgcl  17867  cfinfil  17882  csdfil  17883  supfil  17884  fin1aufil  17921  quotval  20166  fprodser  25232  soseq  25472  sdclem2  26340  fdc  26343  fdc1  26344  rabren3dioph  26770  2nn0ind  26902  zindbi  26903  islindf  27154  bnj62  28795  bnj610  28825  bnj976  28858  bnj106  28949  bnj125  28953  bnj154  28959  bnj155  28960  bnj526  28969  bnj540  28973  bnj591  28992  bnj609  28998  bnj893  29009  bnj1417  29120  lshpkrlem3  29599  hdmap1fval  32284  hdmapfval  32317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-sbc 3126
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