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Theorem sbcieg 3193
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 nfv 1629 . 2  |-  F/ x ps
2 sbcieg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2sbciegf 3192 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   [.wsbc 3161
This theorem is referenced by:  sbcie  3195  ralsng  3846  rexsng  3847  ralrnmpt  5878  fpwwe2lem3  8508  nn1suc  10021  fgcl  17910  cfinfil  17925  csdfil  17926  supfil  17927  fin1aufil  17964  ifeqeqx  24001  2nn0ind  27008  zindbi  27009  trsbc  28625  onfrALTlem5  28628  trsbcVD  28989  onfrALTlem5VD  28997  bnj1452  29421  cdlemk35s  31734  cdlemk39s  31736  cdlemk42  31738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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