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Theorem sbcabel 3354
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 3067 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sbcex2 3327 . . . 4  |-  ( [. A  /  x ]. E. w ( w  =  { y  |  ph }  /\  w  e.  B
)  <->  E. w [. A  /  x ]. ( w  =  { y  | 
ph }  /\  w  e.  B ) )
3 sbcan 3319 . . . . . 6  |-  ( [. A  /  x ]. (
w  =  { y  |  ph }  /\  w  e.  B )  <->  (
[. A  /  x ]. w  =  {
y  |  ph }  /\  [. A  /  x ]. w  e.  B
) )
4 sbcal 3326 . . . . . . . . 9  |-  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y [. A  /  x ]. ( y  e.  w  <->  ph ) )
5 sbcbig 3321 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph ) ) )
6 sbcg 3342 . . . . . . . . . . . 12  |-  ( A  e.  _V  ->  ( [. A  /  x ]. y  e.  w  <->  y  e.  w ) )
76bibi1d 317 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. y  e.  w  <->  [. A  /  x ]. ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
85, 7bitrd 253 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( y  e.  w  <->  ph )  <->  ( y  e.  w  <->  [. A  /  x ]. ph ) ) )
98albidv 1734 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( A. y [. A  /  x ]. ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
104, 9syl5bb 257 . . . . . . . 8  |-  ( A  e.  _V  ->  ( [. A  /  x ]. A. y ( y  e.  w  <->  ph )  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) ) )
11 abeq2 2526 . . . . . . . . 9  |-  ( w  =  { y  | 
ph }  <->  A. y
( y  e.  w  <->  ph ) )
1211sbcbii 3332 . . . . . . . 8  |-  ( [. A  /  x ]. w  =  { y  |  ph } 
<-> 
[. A  /  x ]. A. y ( y  e.  w  <->  ph ) )
13 abeq2 2526 . . . . . . . 8  |-  ( w  =  { y  | 
[. A  /  x ]. ph }  <->  A. y
( y  e.  w  <->  [. A  /  x ]. ph ) )
1410, 12, 133bitr4g 288 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  =  {
y  |  ph }  <->  w  =  { y  | 
[. A  /  x ]. ph } ) )
15 sbcabel.1 . . . . . . . . 9  |-  F/_ x B
1615nfcri 2557 . . . . . . . 8  |-  F/ x  w  e.  B
1716sbcgf 3340 . . . . . . 7  |-  ( A  e.  _V  ->  ( [. A  /  x ]. w  e.  B  <->  w  e.  B ) )
1814, 17anbi12d 709 . . . . . 6  |-  ( A  e.  _V  ->  (
( [. A  /  x ]. w  =  {
y  |  ph }  /\  [. A  /  x ]. w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
193, 18syl5bb 257 . . . . 5  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( w  =  {
y  |  ph }  /\  w  e.  B
)  <->  ( w  =  { y  |  [. A  /  x ]. ph }  /\  w  e.  B
) ) )
2019exbidv 1735 . . . 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, 20syl5bb 257 . . 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 2397 . . . 4  |-  ( { y  |  ph }  e.  B  <->  E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
2322sbcbii 3332 . . 3  |-  ( [. A  /  x ]. {
y  |  ph }  e.  B  <->  [. A  /  x ]. E. w ( w  =  { y  | 
ph }  /\  w  e.  B ) )
24 df-clel 2397 . . 3  |-  ( { y  |  [. A  /  x ]. ph }  e.  B  <->  E. w ( w  =  { y  | 
[. A  /  x ]. ph }  /\  w  e.  B ) )
2521, 23, 243bitr4g 288 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
261, 25syl 17 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  |  ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   F/_wnfc 2550   _Vcvv 3058   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-sbc 3277
This theorem is referenced by:  csbexg  4527
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