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Theorem sbcrexg 3066
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 2994 . 2  |-  ( z  =  A  ->  ( [ z  /  x ] E. y  e.  B  ph  <->  [. A  /  x ]. E. y  e.  B  ph ) )
2 dfsbcq2 2994 . . 3  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
32rexbidv 2564 . 2  |-  ( z  =  A  ->  ( E. y  e.  B  [ z  /  x ] ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
4 nfcv 2419 . . . 4  |-  F/_ x B
5 nfs1v 2045 . . . 4  |-  F/ x [ z  /  x ] ph
64, 5nfrex 2598 . . 3  |-  F/ x E. y  e.  B  [ z  /  x ] ph
7 sbequ12 1860 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87rexbidv 2564 . . 3  |-  ( x  =  z  ->  ( E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 1978 . 2  |-  ( [ z  /  x ] E. y  e.  B  ph  <->  E. y  e.  B  [
z  /  x ] ph )
101, 3, 9vtoclbg 2844 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 1623   [wsb 1629    e. wcel 1684   E.wrex 2544   [.wsbc 2991
This theorem is referenced by:  ac6sfi  7101  rexfiuz  11831  2sbcrex  26864  sbc2rexg  26865  4rexfrabdioph  26879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992
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