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Theorem sbc8g 2771
Description: This is the closest we can get to df-sbc 2765 if we start from dfsbcq 2766 (see its comments) and dfsbcq2 2767. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc8g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )

Proof of Theorem sbc8g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2766 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 eleq1 2100 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
3 df-clab 2027 . . 3  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 equid 1589 . . . 4  |-  y  =  y
5 dfsbcq2 2767 . . . 4  |-  ( y  =  y  ->  ( [ y  /  x ] ph  <->  [. y  /  x ]. ph ) )
64, 5ax-mp 7 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
73, 6bitr2i 174 . 2  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
81, 2, 7vtoclbg 2614 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    e. wcel 1393   [wsb 1645   {cab 2026   [.wsbc 2764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765
This theorem is referenced by:  bj-elssuniab  9930
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