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Theorem sbcreug 2919
Description: Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
sbcreug  |-  ( A  e.  V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
)
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem sbcreug
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2843 . 2  |-  ( z  =  A  ->  ( [ z  /  x ] E! y  e.  B  ph  <->  [. A  /  x ]. E! y  e.  B  ph ) )
2 dfsbcq2 2843 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32reubidv 2550 . 2  |-  ( z  =  A  ->  ( E! y  e.  B  [ z  /  x ] ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
4 nfcv 2228 . . . 4  |-  F/_ x B
5 nfs1v 1863 . . . 4  |-  F/ x [ z  /  x ] ph
64, 5nfreuxy 2541 . . 3  |-  F/ x E! y  e.  B  [ z  /  x ] ph
7 sbequ12 1701 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87reubidv 2550 . . 3  |-  ( x  =  z  ->  ( E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 1721 . 2  |-  ( [ z  /  x ] E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph )
101, 3, 9vtoclbg 2680 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   [wsb 1692   E!wreu 2361   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
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