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Theorem sbcreug 3143
Description: Interchange class substitution and restricted uniqueness 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 3070 . 2  |-  ( z  =  A  ->  ( [ z  /  x ] E! y  e.  B  ph  <->  [. A  /  x ]. E! y  e.  B  ph ) )
2 dfsbcq2 3070 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32reubidv 2800 . 2  |-  ( z  =  A  ->  ( E! y  e.  B  [ z  /  x ] ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
4 nfcv 2494 . . . 4  |-  F/_ x B
5 nfs1v 2111 . . . 4  |-  F/ x [ z  /  x ] ph
64, 5nfreu 2790 . . 3  |-  F/ x E! y  e.  B  [ z  /  x ] ph
7 sbequ12 1924 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87reubidv 2800 . . 3  |-  ( x  =  z  ->  ( E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 2043 . 2  |-  ( [ z  /  x ] E! y  e.  B  ph  <->  E! y  e.  B  [
z  /  x ] ph )
101, 3, 9vtoclbg 2920 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 176    = wceq 1642   [wsb 1648    e. wcel 1710   E!wreu 2621   [.wsbc 3067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-reu 2626  df-v 2866  df-sbc 3068
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