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Theorem sbcex2 3202
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
Assertion
Ref Expression
sbcex2  |-  ( [. 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)

Proof of Theorem sbcex2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcex 3162 . 2  |-  ( [. A  /  y ]. E. x ph  ->  A  e.  _V )
2 sbcex 3162 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32exlimiv 1644 . 2  |-  ( E. x [. A  / 
y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3156 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
5 dfsbcq2 3156 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65exbidv 1636 . . 3  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
7 sbex 2204 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
84, 6, 7vtoclbg 3004 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 343 1  |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    = wceq 1652   [wsb 1658    e. wcel 1725   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  sbcfun  27954  bnj89  29023  bnj985  29261
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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