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Theorem sbcex2 3053
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 3013 . 2  |-  ( [. A  /  y ]. E. x ph  ->  A  e.  _V )
2 sbcex 3013 . . 3  |-  ( [. A  /  y ]. ph  ->  A  e.  _V )
32exlimiv 1624 . 2  |-  ( E. x [. A  / 
y ]. ph  ->  A  e.  _V )
4 dfsbcq2 3007 . . 3  |-  ( z  =  A  ->  ( [ z  /  y ] E. x ph  <->  [. A  / 
y ]. E. x ph ) )
5 dfsbcq2 3007 . . . 4  |-  ( z  =  A  ->  ( [ z  /  y ] ph  <->  [. A  /  y ]. ph ) )
65exbidv 1616 . . 3  |-  ( z  =  A  ->  ( E. x [ z  / 
y ] ph  <->  E. x [. A  /  y ]. ph ) )
7 sbex 2080 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
84, 6, 7vtoclbg 2857 . 2  |-  ( A  e.  _V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
91, 3, 8pm5.21nii 342 1  |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  sbcfun  28090  bnj985  29301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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