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Theorem sbcex2 3155
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 3115 . 2  |-  ( [. A  /  y ]. E. x ph  ->  A  e.  _V )
2 sbcex 3115 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32exlimiv 1641 . 2  |-  ( E. x [. A  / 
y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3109 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
5 dfsbcq2 3109 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65exbidv 1633 . . 3  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
7 sbex 2164 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
84, 6, 7vtoclbg 2957 . 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 1547    = wceq 1649   [wsb 1655    e. wcel 1717   _Vcvv 2901   [.wsbc 3106
This theorem is referenced by:  sbcfun  27657  bnj89  28426  bnj985  28664
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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