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Theorem sbcabel 2920
Description: Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcabel.1  |-  F/_ x B
Assertion
Ref Expression
sbcabel  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( x, y)    V( x, y)

Proof of Theorem sbcabel
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2630 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcexg 2893 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B )  <->  E. w [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
) ) )
3 sbcang 2882 . . . . . 6  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( [. A  /  x ]. w  =  { y  |  ph }  /\  [. A  /  x ]. w  e.  B
) ) )
4 sbcalg 2891 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y [. A  /  x ]. ( y  e.  w  <->  ph ) ) )
5 sbcbig 2885 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph ) ) )
6 sbcg 2908 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( [. A  /  x ]. y  e.  w  <->  y  e.  w ) )
76bibi1d 231 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
85, 7bitrd 186 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
98albidv 1752 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( A. y [. A  /  x ]. ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
104, 9bitrd 186 . . . . . . . 8  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
11 abeq2 2196 . . . . . . . . 9  |-  ( w  =  { y  | 
ph }  <->  A. y
( y  e.  w  <->  ph ) )
1211sbcbii 2898 . . . . . . . 8  |-  ( [. A  /  x ]. w  =  { y  |  ph } 
<-> 
[. A  /  x ]. A. y ( y  e.  w  <->  ph ) )
13 abeq2 2196 . . . . . . . 8  |-  ( w  =  { y  | 
[. A  /  x ]. ph }  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) )
1410, 12, 133bitr4g 221 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  =  {
y  |  ph }  <->  w  =  { y  | 
[. A  /  x ]. ph } ) )
15 sbcabel.1 . . . . . . . . 9  |-  F/_ x B
1615nfcri 2222 . . . . . . . 8  |-  F/ x  w  e.  B
1716sbcgf 2906 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  e.  B  <->  w  e.  B ) )
1814, 17anbi12d 457 . . . . . 6  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. w  =  {
y  |  ph }  /\  [. A  /  x ]. w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
193, 18bitrd 186 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
2019exbidv 1753 . . . 4  |-  ( A  e.  _V  ->  ( E. w [. A  /  x ]. ( w  =  { y  |  ph }  /\  w  e.  B
)  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) ) )
212, 20bitrd 186 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B )  <->  E. w
( w  =  {
y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
22 df-clel 2084 . . . 4  |-  ( { y  |  ph }  e.  B  <->  E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
2322sbcbii 2898 . . 3  |-  ( [. A  /  x ]. {
y  |  ph }  e.  B  <->  [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
24 df-clel 2084 . . 3  |-  ( { y  |  [. A  /  x ]. ph }  e.  B  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) )
2521, 23, 243bitr4g 221 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
261, 25syl 14 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   F/_wnfc 2215   _Vcvv 2619   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by:  csbexga  3967
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