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Theorem sbccsbg 3031
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
Assertion
Ref Expression
sbccsbg  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ {
y  |  ph }
) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)    V( x, y)

Proof of Theorem sbccsbg
StepHypRef Expression
1 abid 2127 . . 3  |-  ( y  e.  { y  | 
ph }  <->  ph )
21sbcbii 2968 . 2  |-  ( [. A  /  x ]. y  e.  { y  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel2g 3023 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  { y  |  ph }  <->  y  e.  [_ A  /  x ]_ { y  |  ph } ) )
42, 3bitr3id 193 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ {
y  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1480   {cab 2125   [.wsbc 2909   [_csb 3003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004
This theorem is referenced by: (None)
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