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Theorem sbc8g 3168
Description: This is the closest we can get to df-sbc 3162 if we start from dfsbcq 3163 (see its comments) and dfsbcq2 3164. (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 3163 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 eleq1 2496 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
3 df-clab 2423 . . 3  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 equid 1688 . . . 4  |-  y  =  y
5 dfsbcq2 3164 . . . 4  |-  ( y  =  y  ->  ( [ y  /  x ] ph  <->  [. y  /  x ]. ph ) )
64, 5ax-mp 8 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
73, 6bitr2i 242 . 2  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
81, 2, 7vtoclbg 3012 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   [wsb 1658    e. wcel 1725   {cab 2422   [.wsbc 3161
This theorem is referenced by:  rusbcALT  27616  bnj984  29323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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