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Theorem sbcalg 2891
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcalg  |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
Distinct variable groups:    x, A    x, y
Allowed substitution hints:    ph( x, y)    A( y)    V( x, y)

Proof of Theorem sbcalg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2843 . 2  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
2 dfsbcq2 2843 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
32albidv 1752 . 2  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
4 sbal 1924 . 2  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
51, 3, 4vtoclbg 2680 1  |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   [wsb 1692   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  sbcabel  2920  sbcssg  3391
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