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Theorem sbc8g 3287
Description: This is the closest we can get to df-sbc 3280 if we start from dfsbcq 3281 (see its comments) and dfsbcq2 3282. (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 3281 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 eleq1 2528 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
3 df-clab 2449 . . 3  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 equid 1866 . . . 4  |-  y  =  y
5 dfsbcq2 3282 . . . 4  |-  ( y  =  y  ->  ( [ y  /  x ] ph  <->  [. y  /  x ]. ph ) )
64, 5ax-mp 5 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
73, 6bitr2i 258 . 2  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
81, 2, 7vtoclbg 3120 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   [wsb 1808    e. wcel 1898   {cab 2448   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280
This theorem is referenced by:  bnj984  29813  rusbcALT  36835
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