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Theorem sbcex2 3330
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)
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 3289 . 2  |-  ( [. A  /  y ]. E. x ph  ->  A  e.  _V )
2 sbcex 3289 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32exlimiv 1787 . 2  |-  ( E. x [. A  / 
y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3282 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
5 dfsbcq2 3282 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65exbidv 1779 . . 3  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
7 sbex 2303 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
84, 6, 7vtoclbg 3120 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 359 1  |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1455   E.wex 1674   [wsb 1808    e. wcel 1898   _Vcvv 3057   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280
This theorem is referenced by:  sbcabel  3357  csbuni  4240  csbxp  4935  csbdm  5048  sbcfung  5624  bnj89  29576  bnj985  29813  csbwrecsg  31773  csboprabg  31776  sbcexf  32398  onfrALTlem5  36952
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