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Theorem sbc8g 2831
Description: This is the closest we can get to df-sbc 2825 if we start from dfsbcq 2826 (see its comments) and dfsbcq2 2827. (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 2826 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 eleq1 2145 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
3 df-clab 2070 . . 3  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 equid 1630 . . . 4  |-  y  =  y
5 dfsbcq2 2827 . . . 4  |-  ( y  =  y  ->  ( [ y  /  x ] ph  <->  [. y  /  x ]. ph ) )
64, 5ax-mp 7 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
73, 6bitr2i 183 . 2  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
81, 2, 7vtoclbg 2668 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1434   [wsb 1687   {cab 2069   [.wsbc 2824
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  bj-elssuniab  10861
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