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Theorem sbcexgOLD 36947
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcex 3288 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcexgOLD  |-  ( 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 sbcexgOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3281 . 2  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
2 dfsbcq2 3281 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
32exbidv 1778 . 2  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
4 sbex 2302 . 2  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
51, 3, 4vtoclbg 3119 1  |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1454   E.wex 1673   [wsb 1807    e. wcel 1897   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058  df-sbc 3279
This theorem is referenced by:  csbunigOLD  37251  csbxpgOLD  37253  csbrngOLD  37256  onfrALTlem5VD  37321  csbxpgVD  37330  csbrngVD  37332  csbunigVD  37334
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