MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbc8g Unicode version

Theorem sbc8g 3011
Description: This is the closest we can get to df-sbc 3005 if we start from dfsbcq 3006 (see its comments) and dfsbcq2 3007. (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 3006 . 2  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
2 eleq1 2356 . 2  |-  ( y  =  A  ->  (
y  e.  { x  |  ph }  <->  A  e.  { x  |  ph }
) )
3 df-clab 2283 . . 3  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 equid 1662 . . . 4  |-  y  =  y
5 dfsbcq2 3007 . . . 4  |-  ( y  =  y  ->  ( [ y  /  x ] ph  <->  [. y  /  x ]. ph ) )
64, 5ax-mp 8 . . 3  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
73, 6bitr2i 241 . 2  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
81, 2, 7vtoclbg 2857 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [wsb 1638    e. wcel 1696   {cab 2282   [.wsbc 3004
This theorem is referenced by:  rusbcALT  27742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator