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Theorem sbcex2 3356
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 3315 . 2  |-  ( [. A  /  y ]. E. x ph  ->  A  e.  _V )
2 sbcex 3315 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32exlimiv 1769 . 2  |-  ( E. x [. A  / 
y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3308 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
5 dfsbcq2 3308 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65exbidv 1761 . . 3  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
7 sbex 2259 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
84, 6, 7vtoclbg 3146 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 354 1  |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437   E.wex 1659   [wsb 1789    e. wcel 1870   _Vcvv 3087   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-sbc 3306
This theorem is referenced by:  sbcabel  3383  csbuni  4250  csbxp  4936  csbdm  5049  sbcfung  5624  bnj89  29315  bnj985  29552  sbcexf  32057  onfrALTlem5  36545
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