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Theorem sbcrexg 3173
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcrexg  |-  ( 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 sbcrexg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3101 . 2  |-  ( z  =  A  ->  ( [ z  /  x ] E. y  e.  B  ph  <->  [. A  /  x ]. E. y  e.  B  ph ) )
2 dfsbcq2 3101 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rexbidv 2664 . 2  |-  ( z  =  A  ->  ( E. y  e.  B  [ z  /  x ] ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
4 nfcv 2517 . . . 4  |-  F/_ x B
5 nfs1v 2133 . . . 4  |-  F/ x [ z  /  x ] ph
64, 5nfrex 2698 . . 3  |-  F/ x E. y  e.  B  [ z  /  x ] ph
7 sbequ12 1933 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87rexbidv 2664 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 2065 . 2  |-  ( [ z  /  x ] E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph )
101, 3, 9vtoclbg 2949 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 177    = wceq 1649   [wsb 1655    e. wcel 1717   E.wrex 2644   [.wsbc 3098
This theorem is referenced by:  ac6sfi  7281  rexfiuz  12072  2sbcrex  26528  sbc2rexg  26529  4rexfrabdioph  26543
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 2362
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 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-rex 2649  df-v 2895  df-sbc 3099
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