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Theorem sbccsbg 3086
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 2165 . . 3  |-  ( y  e.  { y  | 
ph }  <->  ph )
21sbcbii 3022 . 2  |-  ( [. A  /  x ]. y  e.  { y  |  ph } 
<-> 
[. A  /  x ]. ph )
3 sbcel2g 3078 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  { y  |  ph }  <->  y  e.  [_ A  /  x ]_ { y  |  ph } ) )
42, 3bitr3id 194 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ {
y  |  ph }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2148   {cab 2163   [.wsbc 2962   [_csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by: (None)
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