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Theorem sbcexg 3156
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcexg  |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. 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 sbcexg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3109 . 2  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
2 dfsbcq2 3109 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
32exbidv 1633 . 2  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
4 sbex 2164 . 2  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
51, 3, 4vtoclbg 2957 1  |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    = wceq 1649   [wsb 1655    e. wcel 1717   [.wsbc 3106
This theorem is referenced by:  sbcabel  3183  csbunig  3967  csbxpg  4847  csbrng  5056  csbdmg  27652  onfrALTlem5  27973  onfrALTlem5VD  28340  csbxpgVD  28349  csbrngVD  28351  csbunigVD  28353
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-sbc 3107
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