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Theorem sbcalg 2811
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 2767 . 2  |-  ( z  =  A  ->  ( [ z  /  y ] A. x ph  <->  [. A  / 
y ]. A. x ph ) )
2 dfsbcq2 2767 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
32albidv 1705 . 2  |-  ( z  =  A  ->  ( A. x [ z  / 
y ] ph  <->  A. x [. A  /  y ]. ph ) )
4 sbal 1876 . 2  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
51, 3, 4vtoclbg 2614 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 98   A.wal 1241    = wceq 1243    e. wcel 1393   [wsb 1645   [.wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  sbcabel  2839  sbcssg  3330
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